Sang Tao Va Giai Phuong Trinh He Phuong Trinh Bat Phuong Trinh P4

NhUng bai toan tren da cho thay c6 nhieu each de dUa hai vi cua mot phuang trinh ve ham dac triing.. Tuy nhien mot so bai loan kho hdn sc.[r]

B a i t o a n G^a^ he phuang trinh Bai t o a n 24 Gio he ph^Cng IP^f^lU'i^g^X^^ ZlUl [ trinh { ch&ng han 8x^ + x'^ + bx = \/3 - x + x'^ Trong vi phdi, thay X bdi y ta dUdc Sx^ + de phicang trinh + 5x = \/3 - y + y'^ "Tron " x vd y vdo hai vi dU0c "sinh dong" Sx^ + (4) • (! Giai C a c h T a c6 (1) ^ ^ \^4y >| 111 H a m so y ( x ) = l o g i x la h a m ngUdc y y = logi X hdn, chang han ciia h a m / ( x ) = + 5x = \/3 - y + x ^ (1) /1\ y = V i d u 12 Xet mot phu&ng trinh bac ba nao : 8x^ + 6x = VS Ta bien ddi thdnh phuang trinh, trinh = I + + ^Z + g (a) B a i t o a n Gidi he phuang j " , do t h i cua hai h a m doi x i i n g qua difdiig phan giac ciia goc plian t u t h i i nhat y = x B d i vay ( x , y) l a nghiem Thay x bdi y vd thay y bdi x, ta dUOe 8y^ + x^ + 5y = \/3 - x + y^ Ta c6 hdi toan sau t o a n G^d^ he phMng Bai trinh | + ^2 + 5^ = ^ ciia (1) k h i va c h i k h i x = y, nghia la (1) X X H a m so f{t) = 8t^ - 2t^ + At c6 fit) = (1) - 4t + > 0, V i G R nen dong bien t r e n R, d o tvr (1) c6 / ( x ) = / ( y ) , hay y = x T h a y vao he d a cho, ta dUdc = _ X + X^ 8X^ + X = \/3 <^ 4x^ + 3x = (2) Theo bai toan d t r a n g 118, suy r a (2) c6 nghiem d u y nhat "=2 / t>\) (c) ^ t = — ) ^ ;^ - = ^( = ydo = nghich bien n la nghiem d u y nhat ciia he V2'2J /I V i d u Chon phuang trinh (x + y)2 + (x - 1)2 + (y _ i ) ^ / \ < a < ^ (do (1) diioc Tirdo (x;y) = ' v ^ W ^ 3/v/3-v/7 QV ham g nghich bien, do (c) c6 khong qua m o t nghiem T h a y x = - vao — + 5.T; = a^, vdi a = V4; Tren (0; + 0 ) , x e t hai ham so / ( x ) = x, ry(x) = ậ De thay h a m / dong bien ^.T;3 + J ; = - y + x^ - y^ <^8x^ - 2x2 + 4x = 8y^ - 2y2 + 4y x = y = ^ C a c h De (x; y) la nghiem ciia he t h i x > va y > K h o n g m a t t i n h tong quat, gia sii" y > x D a t y = tx, k h i i > T h a y vao (a) dudc Z ^ + ^2 G i a i Dieu kien x G R v a y G R Lay hai phildng t r i n h tr\l nhau, t a dUcJc 8x^ + y^ + 5x - 8y^ - x^ - 5y = | ^^ 2(x'2 + y^ + x y - X - y + 1) = ^ y + X - = y2 + x y + x^ (1) Vay he da cho c6 nghiem d u y nhat biit rdng gidi he doi xiing loai hai, niu lay hai phuang se xudt hien / (x;y) = \ 16 16 16 16 (y-x)/(x,y) = ^ [ ^ ( - - ) = i o / V i d u Ham so y = logi x la ham so ngU0c cua ham so y =^ [ - tren ^dy de tao m.dt he doi xiing loai hai, ta se nhdn hai vi cua (1) vdi (y - x ) , d&n tdi ' (y (0; + 0 ) Do ta co hdi toan sau - .T)(y + X - 1) = (y - x)(y2 + xy + x^) <!=^y2 ~ x"^ + X - y = y^ 244 trinh trie nhau, <^ y^ - y'^ - X + C = x^ - x"^ - y + C Tii day ta se tao rat nhieu he doi xiing loai hai, chang han x^=x^ + y+ C P-S-S' S = = r ^ B a i toan 29 ( D H - 2 A ) ^{y • K h i ?y + r 53 x'^ + xy + - X - ?y -t- f 1= ^ (x + y)^ + ( x - l ) V ( y - i y ^ = ^ ( X i r ^ [ 2/ = ° ( v o ngWem.) V / doi xiJng loai G i a i T a c6 (1) SI l^^^tyf^l «{?(3=is)l2 « 246 ' t ^Il^ai t o a n ^^^^=2 D a t { 1^1^ ;- ; [ {S^ > P ) K l u <16 P = - S''^ - 35 + = i „ ; (^" + ^2/+ ^ = I X + y - x^y - y^'x = - B a i t o a n 31 ( D H S P H N - 0 ) Gidi he phuang C a c b a i t o a n ve h e doi xiJng loai v a c a c b a i t o a n difdc dufa ve he | 22 - P + 3(52 - 2P) - = B a i t o a n Gidi he phuang trinh C a c bai toan r e n luyen v a nang cao B a i t o a n Giai he phuang trinh - 25^ + + 5 + 82 = Vay nghiem cua he l a + \/5 + \/5 , - \/5 - \/5 Vay cac nghiem ciia he l a (0,0), ( — r — , — - — ) , ( — — , — ^ — ) • 4.1.4 ( i + y) = 22 - P + ( - 2P) - = 22 52 - P + = r 53 + xy + x^, t a c6 1= X - (^-'2/^ Dat = y + i , P = yt Ta dudc he I ^^^^1 •r'X y ^ 3y2 _ 9y + y + 3«2 + 3y2 - x){y + x-l) = {y - x){xf + xy + x^) [ y= X y + X - I = y"^ + xy + x^ - X = ^ x(x''^ - x - l ) = ^ x G jo, '{) trinh Hu'dng d S n D a t i = - x He t r d • • K h i y = X thay vao he t a diToc - - 2?/ + = <^ u = i + y= l x^ + y^-x a; - y + y^ - x^ = y^ - x^ Gidi he phuang + 22 = - 3x2 _ Giai he phuang trinh | ^ + ^2 Z ^Js G i a i Lay hai pluWiig t r i n h cua he trii: iihau t a dUdc = P=l Vay ( ; P) = (2; 1), do x va y la n g h i f m cua nghiem ciia he da cho la (x; y) = (1; 1) Tic he (2) nay, neu y = x thi r^ - r^ - r - C = 0, do ta nen chon^ C cho he phuang trinh c6 nghiem "dep" vd bai toan khong qua kho Chang han chon C = thi thdy rang he phuang trinh (2) nhan (0; 0) Idm nghieni vd ta duac hdi toan sau B a i t o a n ( C z e c h A n d Slovak M a t h e m a t i c a l O l y m p i a d 0 ) o trinh f x2 + y2 + xu = : :,U [ x'« + Gidi phuang trinh y4 + x V = 21 Vx + + v'3 - x - v^(x + 1)(3 - x) G i a i Diau kion x G [ - ; 3] D a t ( " = v / p ^ ^ ^ ( V = \/s — X >[j K h i ' / u2 + ?;2 = ^ < - > / ( « + v)^ -2uv = A \u-{-v-uv = \u + v-uv =2 247 ' * i Dat I P > ^ ° ^^^^ - '^^^^ '^^^ [ g=0_2 = P = P = - 5-2 , Vay ( ; P ) = (2; 0) do u va D la nghiem ciia (loai) p a i t o a n ( D e t h i c h i n h t h i J c O l y m p i c / / , I d p 1 ) Gicii fie phUdng trinh ( x + y- v/Sy=_3 Vay G i a i Dieu kien x > - l , y > - l , x y > Phitdng t r i n h thu: hai cua he tirong diWug v i '^^'^^ -^2t = ^ ^ I Z ^ v/r+^ = u = w= X + 7/ + + V ' ( x + l ) ( y + 1) = 16 \/3^ = u = V = • \n = X = — Vay neu dat = x + y, P = x y , dieu kien 5^ > P t h i t a t h u diroe x = x/r+^ = X + y + \ / x y + x + y + = 14 v/3^ = B a i t o a n 33 ( D e t h i c h i n h thiJc O l y m p i c / / , Idp ) Gidi / 5-v/p= \ = 2v/5 + P + l = 14 f P = ( - 3)2 \ + 2^/5 + ( - ) + = 14 (3) Taco he, phuang trlnh +-L] f4= + i l ( ^ + 11 = 18 • ' (3) <^ ^ - 5 + = 14 - ^ { (2) • •';„; - 5 + 10) = - 285 + 196 = • G i a i Dieu kien x / va y 7^ D a t u = u= T h a y vao (1), l a dvtdc ^ + l ) ( i ; + l) = 18 \ + v) {u , {u + vf - 3uu ( u + i;) = I lu + v)(uv + u + v^-l)-=l% (2) (3) r x + y = ^ r y = - x r y = - x ^ / x = lxy = ^ l x ( - x ) = ^ ( x2 - 6x + = ^ ( 2/= 3 y He CO nghiem d u y n h a t (x; y) = (3; ) C a c b a i t o a n v e h e d o i xiJng loai v a c a c b a i t o a n ditdc du'a v e h e doi xiJng loai „, J, 5^ - = 54 - 352 - ^ + 352 + - 63 = B a i t o a n Giai H phuang ( - 3) (52 + + 21) = ^ = trinh rf , Thay vao (b) ditrtc P = 2, thoa man dieu kien 5^ > P Vay { a;y + x2 = + y \y + y2 = + x (1; 1)' (-^; -0 ' (-1 - Do V) (v<5i y G M , t u y y ) ' X - (x; y) = \ ,,,,, , D a p so Nghiem ciia he la u;v} = (2; 1) ! u + Vi = \ = < 26 1225 • K h i = — — , thay vao (2) t a dUdc P = —r , khong thoa m a n dieu kien (a) T i t ( « ) va {b), til CO ^ ^ | G | - y , | • K h i = 6, thay vao (2) t a dUdc P = 9, vay Dat u + v = , uv = P, dieu kien 5^ > P T i i (2), t a t h u diWc r5^'-35P = ^ f P = - \ ( P + + ) = 18 ^ \= - - (6) ^{35^+^85-156 = -: ('••87 = (i;2) 248 Jai t o a n 36 ( D H Q G H N - ) Gidi he 3y = - • * y - 3x = - y B a i t o a n ( D H Q G H N - 9 ) Gidi he phitdng trinh | ylzly^gi B a i t o a n 38 ( D e n g h i O l y m p i c / / 0 ) Gidi phuang trinh gn (0; 1], xet h a m so 5(u) = log2 ( + u ) - loggw K h i (1) ( l c o s ' ' x + 3)'* = c o s x - 768 G i a i Tap xac d i n h R D a t u = 2cosx T h a y vac (1), t a duoc (u^ + ) ' = 4^ ( « - ) , if): (2) fit ^ " g'{u) = (2) Dat V = ^/^{T^ X = sin y = 1 - jg] L^Y (4) trir (5) — D o cos X = - r o (3) > T\i (3) t a c6 he | "4 ij^ ^ * = UQ = ^ ^ ^ J ^ ^ ^ ^ ^ ^ S (0; 1] 3u l n = (1 + 3u) l n cos a; = + 3= 4^/4^ v (1 + 3u)u I n I n " b a n g bien t h i e n v a (4) suy r a cos a; = - uln3 ~ cos De u la nghiem ciia (2) t h i dieu kien la u > - K h i '""'^'^'••''(^^ M U - ( l + 3u)ln2 ( + 3u)hi2 , I ^ K - + gi") smy=Vay c a c n g h i e m c i i a h e d a cho l a thco vc, t a diWc , a,, , - = 4(w - u) {u - v)[{u+v){u^ + w^) + 4] = (6) k2n y = + ^27r V i ?; > va u > - nen t i t (6) t a c6 u = v Thay vao (4) t a diWc ' X = arccos ^ + n2iT X = - arccos ^ + n27r ^ y = arcsm - + m2n y = arcsin - + m27r o o 1 X = arccos - + n27r X = - arccos - + n27r X = - 4?/, + = {u - l)2(w,2 + K + 3) = Vay (1) <^ c o s x = <f=^ cosx = i y = IT - arcsm - + o • y = TT — arcsin - + m2n o 7i = m2n x = ± ^ + fc27r, A; G Z B a i t o a n ( H S G T p H o C h i M i n h , n a m h o c 0 - 0 ) Gidi he ^ log2 (1 + c o s x ) + log3 ( c o s i ) = log2 (1 + 3siri2/) + logg ( s m y ) Tren khoang (0; + 0 ) , xet ham so f{t) ^ ' ^ ^ " ^ ^\ V i ^+ \2 + i = ^^^y { W = G i a i Dieu kien | y ^ G i a i Dieu kion cosx > 0, s i i i y > Lay (1) t n r (2) then v6, t a diMo i B a i t o a n Gidi he phuang trinh x^ khong la nghiem ciia he phiTdng t r i n h da cho Lay (1) trir (2), t a duoc (3) \/x2 + 91 - Vy^ToI = ^ y - - = l o g a l l + 3f.) + loggf K h i ^ (x - y) Vay ham so / dong bien t r e n khoang (0; + 0 ) T t f (3) t a c6 •^x = / (cos x) - f (sin y) <^ cos x = sin y T h a y vao (1) t a ditdc y ^ + 2/^ - y-x x^-y^ v / y ^ + V x - s/x^ + 91 + ^y^ + 91 x +• y v/x2 + 91 + x/y2 + 91 ^ v^r=^ + x + y v^x2 + 91 + + (y - x) (y + x) ^Jy^ Vx - + x + y + 91 ^ v ^ ^ + v/^^ (4) v/x2 + 91 = v ^ x - + x^ ^50 251 = + x + y > Thay x = y vao mot t r o n g hai phitong t r i n h cua h?, t a ditdc l o g ( l + c o s x ) - l o g ( c o s x ) = (2) <(^\/x2 + 91 - ^ 10 = - + - ^ v/x-^ + 91 + 10 - ~ ^ + \ / ^ ^ + l 4.2 4.2.1 x + = X + + V ^ ^ + 1' v/x2 + 91 + 10 > 5, t r o n g k h i V6i T > t h i x + + \/x^+ + 91 > X Vay (3) ^ x + 91 + 10 > X + H e CO chiia mot nhat) bac hai = (3) I r,« '2M I - - X - 3j X = H e C O yeu to dang cap phu-dng t r i n h d S n g c a p he co chi'la m o t phudng t r i n h dang cap bac hai (khong chiia cac g6 hang bac nhat va t u do) t h i ta xet cac trirdng h^Jp , TrircJng hdp : x = T l i a y vao ho t i m cac nghiem dang (0; y) neu c6 tlniSn nhat de t i m t Sau t i m x va y , ^ <1 Bai t o a n 42 ( H V N H - 0 ) B a i t o a n 41 ( D e t h i c h i n h thiJc O l y m p i c / / 0 ) Giai he phuang trinh / x ( l + 3y) = \(y3 _ 1) = / x2 - 2xy + 3y2 = \13xy + 15y^ = Gidi he If.: r 15 ( ' y ) ^ , / + = Vx/ x ^ y x = 5y X y 2^- X Khi X = 5y, thay vao (1) t a diTdc y^ = i (*)^ (x; y) = Lay (1) trfr (2) thoo vo, t a dildc y){e + ty+ y'^ + 2>) =Q ^ \ t = y •^"i - = = D a t i = 2u, t h a y vao (3) t a dildc 8u^ - 6u = 4u^ - 3'u = cos - (4) Theo bai toan d t r a n g 119, suy cac nghiem cua (4) la cos—, cos-g-i c o s - ^ Vay t a t ca cac nghiem cua (3) la c o s ^ , c o s - ^ , c o s - - ^ Nghiem y y y y cua (*) la ' X = — TT cosy = 2cos- X = cos 57r 2/ = cos 252 X = cos 7n y = cos ' , {x; y) = \ - 2.- + 3.- = ^ x-^ (3) 7r % chia ca hai ve cua (1) cho x^ ta dudc T h a y vao (1), t a diTdc ^-2,1 y = ± — Vay X t^-y^-3{y-t)=0^{t- (1) (2) Giai Ngu X = t h i t h a y vao (2) dudc y = 0, thay vao (1) thay khong thoa man Xet X ^ C h i a ca hai ve ciia (2) cho x^ t a dUdc G i a i Dg thay x = khong thoa man ( * ) T i e p theo xet x 7^ K h i ~ } thu4n , Trirdng hdp : x 7^ D a t y = tx (hay t = ^ ) , thay vao phiTdng t r i n h + 91 + 10 X = He phiTdng t r i n h c6 nghiem d u y nhat x = y = Dat t = ^ , t a dUdc | ^3 (tiJc l a x^ = X = ±3 nem {x; y) = (3; 2), (x; y) = ( - ; - ) He da cho c6 b6n nghie: K A\/2 ( s/2\ (3; 2), ( - ; - ) 5\/2 Qai t o a n Gidi he phUdng tnnh inh {xl-2xy-2y^ = Q \2 + y2 ^ 2x + 3y = 19 ^• J ^ap s6 / I 14 X =: J/= ' 57 f y = - 10 ( l-3v/l7 -l+3\/l7 253 X = ; < ' 2/ = l + 3vT7 -1 - \ / l 4.2.2 H e C O h a i phrfdng t r i n h b a n d a n g cap bac h a i a^x"^ + b-ixy + C2y^ = , Khi = 16, he (A) c6 nghiem, chfing han (x; y) = (0; s/U) ^, , I<hi nL i=- 16 He [A) c6 nghiem k h i va chi k h i (3) c6 nghiem ( v i neu c6 i j-jii CO X cho b6i (2) va c6 y tilt y = i x ) , nghia la d-^ Dua ve dang phttCng t r i n h da xet muc 4.2.1 T i l hai phudng t r i n h ban dan^ cap bac hai nay, t a tao mot phitdng t r i n h dang cap bac hai n h u sau dx {aax^ + h'lxy + = d-^ (aix^ + bixy + ciy^) C2y'^) B a i t o a n 44 ( D H Q G T p H o C h i M i n h - 9 ) Cho he phMng trinh CO (A) y^y tic hai todn 44 ta thu diMc kit qua sau: Neu edc s6 thuc x, y thod fiian dieu kien 3x^ + 2xy + y^ = 11 thi bieu thiic x^ + 2xy + 3y^ cd gid tri nho nhat /d 22 - l l \ / m gid tri Idn nhat /d 22 + l l y ^ Bai t o a n 45 Cho cdc so thUc x,y thod man dieu kien x ^ - x y + y^ = gid tri Idn nhat vd gid tri nho nhat cua bieu thiic G = x^ + x y - 2y^ ^ = 17 + m He (A) c6 nghiem k h i va chi k h i 11.3 = 17 + m Hu'd'ng d a n Goi T la tap gia t r i ciia G Ta com CO nghiem: m = 16 I = 11 x2 + t x + 3i2x2 = 17 + m ^ x2 (3 + 2t + t2) ^ II { x ( l + 2t + 3f2) = 17 + m (1) (2) l + 2t + 3^2 11 + i + /2 ^ max G = - + 2\/7, m i n G = - - ( m - 16)f2 + 2(m + 6)/- + m + 40 = (3) a) K h i m = 0, tir (3) c6 -laf + 12i + 40 = <^ 4^2 - 3t - 10 = • K h i /, = 2, thay vao (1) t a dUdc l l x ^ = 11 ^ (x; y) = (-1; - ) la nghiem ciia he ' = -4- x = ± Vay ( x ; y ) = (1;2) (A) • K h i i = - , thay vao (1) dUdc ^ x ^ = l l < ^ x = ^ < ^ x = / \ ^4v/3 5v/3\ , / 4v/3 5v/3\ la nghiem ciia he , (x; y) = Vay (x; y) = ^ , ^ K h i / f t = he (/I) CO bon nghiem (l;2),(-l;-2), ^> Tien hanh t i M n g t u nhir bai toan 44 t a suy dudc ket qua : Ho (*) c6 nghiem k h i va chi k h i - - ^ < m < - + 2%/7 Vay Lay (2) chia (1) thco ve t a ditdc 17 + in 4v/3 5N/3' ^ 4\/3 ' ~ ' 254 Tim k h i va chi k h i he sau f x - x y + y2 = \2 + xy - y = m T i e p theo xet x 7^ Dat y = tx T h a y vao he {A) dvtdc f 3x^ + 2tx^ + t'x' < m < + 11 v ^ - l l \ / < m < + l l v ^ <^ 22 - l l v ^ < 17 + m < 22 + I V a) Gidi he m = b) Tim m di he c6 nghiem 3y + 10m + 338 > <^ - 11 j ^ l t luan : He {A) c6 nghiem k h i va chi k h i - l l \ / < m < + l l \ / C h u y 1- 7a 3x'^ + 2xy + y^ = 11 + 2xy + 3y2 = 17 + m l i a i Neu X = t h i | A ' = -rr? 2\/7 L i f t i y K h i t i m gia t r i Idn nhat, gia t r i nho nhat bang phUdng phap tap gia tri, ta khong can chi ro gia t r j ciia bioii so biou thiic dat gia t r i Idn nhat, gia t r i nho nhat , Bai t o a n 46 ( X? -\- xy I X y Gidi he phMng trinh < iJap s6 ( x ; y ) - ( ; l ) , - y^ = (x; y) = ( - ; - ) ^^•2.3 H e d a n g c a p b a c h a i aix"^ + bixy + ay^ + di ^ a2X^ + b^xy + C2y'^ + d2 ^ 5^3^ 255 0 xy' G i a i Vai y = t h i he t r d | ca hai ve ciia (1) cho ( v o nghiem^ X e t y 7^ Chi;i ^^-f^-^ va dat - = i , t a dUdc ,^ y^{2e-U • Neu - 1) (2t - 1) < i ^lf,±l = ^ ( {x + y)^ + y^<Q ^/^= j { | t luan : He da cho c6 nghiem nhat k h i va chi k h i a = |; ^ + l) = l ^ y \ t ~ \ ) { t - l ) (x; y) = {s/a; 0) la nghiem ciia h^ Suy ngu a > t h i h § c6 i t nhat hai pghiem X e t a = 0, t a c6 he j, (3) t ^ - t < t + l ^ t ^ - t - l < Q ^ \ - y / < t < \ y/2 T h a y x = ty vao (2) t a diWc I = l (4) < t < t h i (4) vo nghiem Suy he vo [ gai t o a n 50 Tim a de he sau c6 nghiem 5x2 ^ , ^ + 2j/2 > O < 2a - : < -.^2 _ G i a i He da cho viet lai nghiem 1\ • X e t ( i - 1) (2« - 1) > <^ < € f -^.^'-^xy-ly^<-l U (1; + 0 ) K e t hdp vdi (3) ta c6 2; dieu kien ciia - \/2; i\&t^ 7x2 - x y + / < ( ) 2a + V U ( ; + ^ ] K h i (4) cho t a nghiem : (1) Cong (1) va (2) theo ve t a dUdc y = ±^ 16 (i-l)(2t-l) —X (^-l)(2i-l)' 16 -6 - —xy " + -y " -< a + 't +• \ vdi i G <^16x^ - 16xy + 4y^ < 1\ 1-v^;U ( l ; l + V2 2a+ ^ (4x - 2yy ' < 2o + Vay he da cho tUdng dUdng vdi X? - 2xy < a x'^+ 2xy-'2y'^ <2a+l B a i t o a n C/io he a) Chiing minh rhng vdi moi a he da cho ludn c6 b) Tim a de he c6 nghiem ( ^ ' r x + x y + 2y2 > ' (1) nghiem nhat Tit (3) suy 2a + < a < T a xet he Giai r a) K h i X = 0, h f t r d t h a n h | ~J2y^<'2 a + |2a + l | {x;y) <^ ^ 2/2 > - a > 2a + 1 Suy 5x2 _^ + 2j/2 = ^ (4x - 2yf = ( y = 2x I 21x2 ^ b) T i r kgt cjua cau a) suy : K h o n g ton t a i a de he (*) c6 nghiem d u y nhat Tigp theo t a chiing m i n h , vdi a < B a i t o a n 49 Xnc dinh cnc gin tri ciln a de he snu c.6 nghiem (a^o; ijo) la nghiem ciia he ( H ) , k h i r x'^ + 2xy + 2f < a \ Axy -y^ <a (2) nhat + y"^ < a Vay neu a < t h i h? da cho vo nghiem T i e p theo xet a > N h a n xet rang k h i (x; y) = (0; 0) t h i he ( I ) c6 nghiem T h a t vay, goi | ( - „ ) ^ = o < ^ , ^ % (xo; yo) la nghiem ciia ( I ) , suy he ( I ) c6 nghiem Vay he da cho c6 'Nghiem k h i va chi k h i a < - - 4b 256 (11) r 5xg + 4xoyo + 2y^ = (1) H G i a i B a t phUdng t r i n h (1) v i l t lai (x + yf ^ ^ ' ^ < (^•; y) la nghiem ciia (*) vdi m o i a = (0; 0, vdi ( > max {x; y) 257 f a can xac d m h cac gia t r i ciia — (*) c6 nghiem, ti'rc la phirong t r i n h L t f u y D i g m mau chot cua Idi giai la tim dufdc bat phvtdng trinh (3) Vj^y (3) dUUc tim l a nliU the nao ? Vdi m < 0, ta c6 { (!!l^l)t^+(- 5mx^ + Amxy + 2rm/^ < m o ( m + 7)x^ + ( m -4)xy <^7 77i< o + (2m + 2) + —-5 2a — < 3m + (—Y \ ' Vay m < ^ ^ = 0c6 /m Cong hai bat phudng trinh cua he, ta dUdc T a can chon m oho + At 36 f — ) + < <^ 61 < — < 62, ^^1,2 Va / a < y ( - 62) x^ + x y + 2y2 = ^ < y= ±- 62 + (nhan) l2 [2(1-62)] 2) m = - - 2^74 suy gia t r i Idn nhat ciia m la 62, dat dUdc k h i va chi 2v/5m + 7.V2m + = ± (4m - 4) ( m + 7) ( m + 1) = ( m - 1)^ 18 + 62+4 X bang binh phUdng cua mot bieu thiic nao M u o n vay thi (loai) = 62+4 {V5m + 7x)^ + ( m - 4) x y + (\/2m + 2y)^ m = -5 - ^ n^en \m 0, m + > 0, 2m + > va = » \ / m , + V m + = ± (2m - nghiem V i - 4.2.4 62 + 2(1-62) + H e dang cap bo phan Vay t a se nhan bat phiWng t r i n h t h i i nhat cua he vdi - ^ , sau cong v6i K h i gap he dang cap bo phan, t a thudng sii dung phep the de tao he dang cap Nhieu k h i da biet chfic dua dUdc ve he dang cap t h i t a lai khong lam dieu m a giai luon bKng each dat x = ty hoftc y = tx (2) va thu ditdc (3) B a i t o a n 53 Giai he phuang trinh o r 3J:2 + 10x?y - 5?;^ < - Tim a dd he sau c6 nghicm : < 2,0 ^ - a +zxy ^2/^1^^- B a i t o a n 51 D a p so a < G i a i Ho phUdng trinh da cho vict lai + y3 = x2y + 2xy2 + y3 = -1 • Khi y B a i t o a n 52 Xdc dirih gid tri l&n nhdt cila m dc hC aaii c6 iighiern 7^ X 2) X e t x^ + x y + 2y^ > Dat x^ + x y + 2y2 = a, < a < K h i m _ x^ - 4xy + 5y^ a x"^ + xy + 2y2 771 b) N6u y ^ t h i m _ ~ Dat - = t, thay vao (4) diroc 2f y Ngu /, = t a CO he { f ^ f N6u t = -\a C O he { f^fy Neui = i t a c h e { a) Neu y = t h i — = 4^ m = o - 4t + t2_,.i + ' 258 ^ _ X ~ y' Vay he C O nghiem la i (2) (1) (3) f - ] " - f - ) +1-0 \yJ \yJ ' - V \vJ Giai t h i x2 + xy + 2y^=0, \ + y^ - x^y - 2xy2 = .7:3+y3 = l / ^ x^ + y-^ ^ T a co x + xy + 2y2 < \ Axy + 5y2 = m 1) Ngu { J = ^2 + y^ = t^y + 2xy2 + y3 = | = " ^ - f - 2t + I = <^ t = ±1, O .T ^ = y = { x i f^£'='^ / (4) -y ' 2^^ \ ' 259 s/2- nghiem) - B a ii t o a n 54 Gidi he phuang trinh { "^^'^^ = + Vs/v •dix = y t a duoc { ^2 G i a i Dion kion x > 0, y > • X e t 2/ = 0, khong thoa m a i l he phvtdng t r i n h • X e t y T^O, dat = t^, dieu kien i > va t 7^ He t r d t^y-8 = t + y ^1 - 1) = tifo «•/ i T&co ^ ^ \ ^ i X = 2y duoc: { - r , ^ (do (1;1), dUdc t = - Vay y = -g 3= =4- = = (-1;-1) 2A/- B a i t o a n 56 ( D e n g h i O L Y M P I C / / 0 ) '^ trinh Giai • Xet y = T h a y vao he t a dUdc kien t a duoc nghiem ciia he phiTdng t r i n h la (x; y) = (9; 4) „ ™^ Gidi he^phudng ^ + 8y^ - 4xy2 = 2x4 ^ 8^^4 _ 22; - y = x = K e t hop vdi dieu B a i t o a n 55 ( D H - 1 A ) Gidi he phuang trinh r x y - x / + y - ( a : + y) = (1) " = 1) /2 ^ 3, - 8i^ + t + = Q ^ t e { , - - , - } D o i chieu dieu kien ta {*) ^ Cac nghiem cua he la ^23T~^ = ^ ' +«2 _T ^ i - l ^2 ^ (x; y) = ( ± ; ± ) ^ { y2 ^2 vay (.T; y) = (1; 0) la nghiem ciia he da cho • Xet y ^ D a t x = ty K h i « y3(t^ - 4( + 8) = G i a i Ta c6 i/(2t'' + ) = i + (do y 7^ 0) 2*^/ + 8y4 - 2fy - y = (2) ^ xyix"^ + y2) + = + 2/^ + 2xy - 4i + 2/4 + + y'^){xy - 1) - 2(xj/ - 1) = xy = <=^(xy-l)(x2 + y _ ) = 0<^ x^ + y- - ^ •^i^ - Tru'dng h d p :?; = - , thay vao (1) t a diWc X x - -X + — - x - - = < S ^ x - - + ^ = X <^3x4 - 6x2 + = <^ (x^ - 1)^ = x^ = <j=! X = ± 2/4-1 ^ 2/4 + / - 8/2 ^- 4/ + 10/ + = 2/4 + + 12/ = <^ / e { , , 6} hi / = t a CO (x; y) = ( 0; K h i / = t a c6 (x; y) = ^ (x; y) = ( ; 0), (x; y) = K h i / = t a c6 (x; y) = ^ ; ^ ) \ He CO bon nghiem (o; ^) , (x; y) = (l; , (x; y) - ^ ) Vay (x; y) = (1; 1), (x; y) = (-1; - ) la nghiem ciia he = T h a y vao (1) t a dudc Tru-dng h d p : x^ + B a i t o a n 57 Gtwi, he phuang trinh 5x2y - 4xy2 + 3y^ - (x^ + y^){x + y) = <i=>5x2y - 4xy2 + 3y^ - x^ - y^ - x^y - xy^ = <;^4x2y - 5xy2 + 2y^ - x^ = G i a i Xet y = 5X y „ X^ +2 ^=0<^ yi 260 x = 2y X = y L y '^y^^^^Jzl^ x = Xet y 7^ Dat / = - <^ x ty He da c l (3) Neu y = t h i tiJt (3) suy x = 0, m a u thuan vdi x^ + y2 ^ 2, vay tiep theo t a chi xet y 7^ K h i chia ca hai ve ciia (3) cho y^ t a diWc 4.T;2 ( < » ^ yi | Chia (1) cho (2) vc theo ve t a duoc ^^_:±±l = i z i ^ / _ / _ / /2 + / - / - / = -1 + = 0.= '=3- 261 H e d a c l i o c6 b o n n g h i e m ( ; ) , (1; 1), ( - ; 1), B a i t o a n 60 ( H S G Q u o c gia n a m hoc 2006-2007) trinh 3^3_ Gidi he phMng B a i t o a n 58 (1) ^ X + iJ trinh x'+y' - 12 y + Gidi he phudng v/5 = ' 3x (2) = l I \ G i a i N h i n v a c he t a de d a n g n g h i d e n v i e c t h e so t i t p h u d n g t r i n h (2) vao i ) h \ m n g t r i n h (1) dfi r o phirclng t r i n h t h u a n n h a t ( d a n g c a p ) D i o n kioii CO (2) 4=> (x2 + y2)2 x + yj^O.T^ G i a i D i ^ i k i e n x > y > 0, y + 3x ^ T h a y x = 0, y = vao he t h a y k h o n g t h o a m a n V a y lie p h i f d n g t r i n h t i t d n g d i t d n g = T h vao (1) t a d u d c : + y)={x' i3x'-y')ix ^ S x - i -y' + 3x^y y'f - xy^ = x* + y'+ + 'Sx^y - 2x^y^ ^2x* + - + 2x^y\ - xy^ - 2^"* = X = y + 'Sx \ sjx] •(1) /y -^1 -2y 2x2 ^ ^ Q X y -12 y + (y - 9.x) (y + x ) + 12.xy = 3.r y = • K h i 2x2 y = 4- y2 ^ t a c6 x ' + x y + y2 = <^ x^ + ( x J 1^ = <f=> x == yy = 0, + ^ k h o n g t h o a x^ + \f = I T n t n g h d p n a y l o a i • Vdi X - 1 y t a c6 2x^ = \ = ^;2 vay x = y = V a i x = - y tac6y^ = -.Vayy = - ^ , x = - ^ , y - = ^ : , ,X =- ^ K e t I r a n : H e c6 n g h i e m ( x ; y ) l a f-2 .7!' 7!;' B a i t o a n 59 (2) y + 3x yfy N h a n (1) v a (2) t h e o ^•e t a d u d c px = y <^ ( x - y) ( x + 2y) {2x^ + xy + y^) = <^ 12 y + 3x 12 - Gidi cdc he phMng ' innh ( P + 4yy - ?y^ - 16x = | y = 5x2 ++ 7i;' Vv/2' ^j 4= +\/3x -i= = - ^ V-i' + V3 = v^^x B a i t o a n 61 ( D e t h i c l n ' n h thu-c O L Y M P I C phiidnq trinh jj;, , r x^ - x = D a p so a) H e CO n g h i e m ( x ; y ) l a (0; ) ; ( ; - ) ; ( ; - ) ; ( - ; 3) + 2y ^ ^- - + - he 2 x + y X + y/ | = 4v^ B a i t o a n ( D e n g h i O L Y M P I C / / 0 ) Gidi he phiMng V4 Gidi ' ' y-M2x, b ) H e CO n g h i e m ( x ; y ) l a x + y ; ^ + ^ j x + y 262 2^3 30/04/2000) P a i t o a n ( H S G q u o c g i a - 9 ) Gidi he phtidng Lrinfi Ix2-3y2=6 (3;l);(-3;-l); = + K e t h d p d i e u k i o n ,suy r a he c6 n g h i e m d u y n h a t | ^ I I ^ ( + ^ x / ) + \/ sau : ,, D g ( x ; y) l a n g h i e m c i i a he t h i x > 0, y > 0, d o y = x T h a y vao (1) d i t d t ^3- 3x -9x 263 j = Innh Giai Dieu kien x > 0, y > 0, x + y =>;^ De t h i y x = 0, y = khong thoa man he, vay chi can xet x > va j / > Dat u = > 0, v = ^ > Khi 2u + v 2ii + i ' + v 2u + i ; u'^ + 2u ^ .^18 + 9(1),3 = - 0/ ? ; \\ 30 g ) - 1 (1) Vdi y = tx thi phirong trinh thii nhat ciia he trd ' (2) 2.x3 + ^3.^3 ^ JO ^ ^ 2u + t; 2iL + v 4v - u u V v?' + z;-^ uv + i;^ <^4t;u^ + 4i;^ - uv^ = 2u^v + uw^ - 2uu2 = Vay + -L (2uu^ - u^) + (4^^ - 2uv^) = u = 2t) = 2(v/2 + l ) ^ = ( + 2^/2) = A^^ r u = 8(3 + 2v/2) \?; = 4(3 + 2v^) 10 • V 2+ • + V i du V ^ i { J ^ to CO I /2±3N/2\ ^4 ~ ^ ^ ^ ~_~3^ ^ Q ^^'^^ phudng phdp titdng til nhu giai he co yeu to ddng cap, ta can xdy dung phudng trinh th:ti hai cua he cho no khong chiia so hang tU Ta c6 bdi todn sau f X = 64(17+12N/5) y = 16(17+12^) + ^ 10 (x;y) = ( l ; ) , (x; y) = Bai toan 65 Giai he phUdng trinh He CO nghiem nhdt { ^ I X = / = 2?' vao (1) ta (hruc -|= ^ Vay ling vc3i ba nghiem t tim difdc d tren thi he da cho c6 ba nghiem la ^u^{2v - u) + 2u2(2u - u) <t4- (u^ + 2v^){2v - u) = Thay 10 + t^ ^2vu^ + 4w^ - t= 9t^ - 30t^ -f lot + 28 = <^ ( i - 2)(9t^ - 12< - 14) = y/v Nhan (1) va (2) theo ve ta diWc j'^^'V Dat t = - , _ J _ _ 4u + 2u + ^^^ISx'^ + 9y^ = -lOx^'y + SOxy'' - lOx"* | ^ ^ ^4 + J_~3y ^ Q v|) Hirdtng dan Khi y = 0, he da cho trd | ^ ^ ~1 Q <^ X = - Vay 4.2.5 nghiem co dang (x;0) ciia he la (-1;0) Tiep theo xet y ^ Dat x = ty, thay vao ho ta dudc Phufdng phap sang tac bai toan mdi V i du V6i { J = to CO I + ^ giai he, ta se di(a ve mot phMng _9 tnnh hac ba theo i — hon niia phuong X 2x'^todn + 2/3sau ^ 10 trinh c6 "nghiem, dep" t = Ta c6 bdi x^y - 3xy2 + x^ = - Bai toan 64 Giai he phuang trinh Giai Khi x = , he da cho tr6 | g _r_^g = Vay he khong c6 nghiem dang ( ; y) Tiep theo xet x 7^ TiJt he da cho, ta co {2x^ + y3) = _ 10 (x^y - Sxy^ + x^) 264 ty^ 2/ = 2x, nen tV - 2iy.y2 = - + y* + ty-Sy =0 <io^o J ((^ _ 2t) = - y3 ( ( + Suy -1 - 2t ^3t^ + 1) = _ t 3-t /4 +1 <=> -^t'' - l = 3t^ -t* - 6t + 2t - 6« + = ^ (i - 1) (3*2 + i - l ) = ^ i € | 1, - ^ ^ ^ Lifu y Ta biet chac ch^n rang se co ( = la vdi x = y = thi < = - = 1y 265 C h u y Viec sang tdc cdc he phUdng trinh duac gidi bang each dit.a ve cdc phicang trinh dang cap bdc hai nhic cdc hai todn 60, , 62 se duac trinh bay a muc 4.6.2 d trang 311 Sau day ta se tiinh bay mot cdcli sung tdc klidc cuug khd nhanh chong cho cdc he phiMng trinh dang {xem vi du ) at x^ = i ta ditdc he D V i d u Xct , = (2 y = 4.T + v^y Khi (1 + —KA V y + oxj y + 5x = ^ ^ = 9x y + bx 77 + 5x7 X + 4= = + vAr - = - =^ + _ ^ y + 5x = l + a^ I 2a = (1 + a'){2 2\ + 2a - + 2a) = (1 + a'^){Aa + 1) l + a^ +2a + fv2 r-'l; , 1+^2 11 = (l + a ) 11 X' + - \/3 = M d i khdc - 2a +2a + 4a - 11 + 22a = 26a - * ' !? -aJ VI DX 7^ 0, Va nen neu 4a 4- = t h i D = 0, he v6 nghiem T i e p theo t a xet a \ K h i y + 5xJ C (1 + f v ^ ) z + (1 - 2fv)x = (1 + a'-^)^ + 2(1 + a ) x = 11 Ta tfnh cac dinh th^ic y = ax, x^ = z 26a - Or D Ta CO bai todn sau 4a+r ' D (l + a ) ( a + l ) ' Dieu kien x!^ = z cho ta phUdng t r i n h B a i t o a n 66 Gidi he phiMng trinh 81 26a - (4a+1)2 (l + a ) ( a + l ) <^ 81(1 + a^) = (26a - 7)(4a + 1) v a = 44 a = -23' <^81a2 + 81 = 104a2 - 2a - 4* 23a2 - 2a - 88 = 4.3 He bac hai tong quat 4.3.1 e, V6i a = 2, t a diMc { J = N h a n dang v a phifdng p h a p giai X He bac hai vcli hai an x va y la | • • \ ^^^^J + b2xy '''A t w'^" t + C2y'^+ d2X + C2y ^ (*) h- Mot so trirSng hop dac biet (doi xi'mg loai 1, loai 2, dS,ng cap ) da difdc xet (1 cac phan tntdc K h i cac t m l i chat dac biet khong t h i he (*) d\trtc giai theo mot so chnng se dudc t r i n h bai cac bai toan 67 d trang 2GG, bai toan 68 trang 267 T i i y nhien phirong phap khong i)hai la t o i uu N h i n chung c:ac dang thitcJng gap dcu dita t r c n nigt vai dac t h u cua dang bfic hai Nen biet khai thac cac tfnh chat dac biet ta se t i m ditrJc Idi giai ngiln ggn B a i t o a n 67 Gidi he phitcing trinh + y + a; - 2y = • ^ + y ^ + 2(x + y) = 11 {5 Giai = 44 Vdi a = - — , t a dildc 23' / y = 153 ~ 44 - 44\ \4 23 17 23 17 17' Vay he da cho c6 hai nghieui (x; y) = (1; ) , (x; y) = J^! J^)" L i f u y Bai toan 67 se dUdc giai nhanh hdn neu ta n h i n thay dUdc : L a j ' hai phitdng t r i n h cua he trir nhau, t a se t h u dUdc mot phitdng t r i n h bac nhat theo hai an x va y, tit day r i i t y theo x, giai bang phifdng phap the 4.3.2 S a n g tac cac he bac hai tong quat '' = Khi { ""1 + y' " + ^y = - [ x ' ' - x y + y ' ' + X - 2y = 1^Vay ta thu ditOc mot he bac hai tong quat, he ch&c ch&n c6 mot nghiem "dep" la {x;y) = {3;0) Ta CO bai todn sau day , V i d u Xet hai s6x = 3vdy K h i X = t h i he viet lai | _,_ 2y - 11 "SliiC-ni K h i X 7^ Dat y = a x , thay vao he da cho t a ditOc 4.(i_2rk)x = ( l + a ^ ) x ' ^ + ( l + a ) x = 11 (l+a2):;;2 x + ft'-^x^ + X - 2ax = I x^ + d^x^ + 2x + 2ax = 11 2GC B a i t o a n 68 Gidi h$phmng trinh { ""l^~ H'[ X — xy + y + X — ,o Zy — i^- ^ / x2 + 22/2 + a;y + x - ? / = - ^ \2 - 2/2 - x y + 15x + y = - Giai • K h i T = t h i he viet lai | ^2 ^ 2y = 12^ "^^ nghiem fa thu dUdc bdi todn • K h i X 7^ D a t y = ax, thay vao he da cho t a dilOc •+ (1 + a ) x + ( Q - ) i = - ( l - a + a2)x2+(l-2a)x=12 Dat .?• sau B a i t o a n Giai he phudng trinh ^ • ^ [ 2y2 + x y + x - lOy = - \2 - y2 - x y + + 4y = _ Htfdng d a n D a t x = w, - va y = ?; + 3, t a dUdc he dang cap bac hai = z t a dvTdc he (1 + Q ) + ( Q - ) X = - {l-a + a^)z+{l-2a)x=12 y = ax, x^ — z I 3u2 -uv-v^ + ^2 1- a + 2a - _ 2Q 2(Q-2) - 2a + a2 l - a + a^ = - a ^ + 7a2 - 8a + + 2n.u + a2 + 2t;2 ^ 4^,; + 2^2 + uv + bu + av + ab + = - a + 45 -3 12 = \ 'Tf^ >' LuM y- Phep d f t t x = u - v a y = i; + dUdc t i m r a n h u sau : Ta dat j ; = u + a va y = t; + 6, vdi a, t i m s a u K h i do, t h a y vao phitdng t r i n h thit nhat c i i a ho, t a d u d c T a t i n h cac d i n h t h i i c -3 12 , ^ M + a - 10?; - 106 = -12 De t h u dutdc phUdng t r i n h dang cap bac hai t h i dieu k i e n la = ISa^ - 3a + 15 ^1 / 2a + + = \6 + a - = K h i D = 0, tufc la a = t h i he v6 nghiem T i e p theo chi xet a ^ D i o u kicu x^ = z cho t a p h i M n g t r i n h de xac d i n h a r ^ a = = -2 Vay ta dat X = u - va y = ?; + Ngoai t a c6 the lam nhanh hdn nhiT sau: L a n lUdt dao h a m hai v c phUdiig t r i n h t h i i n h a t thco b i c n x ( x e n i y la X z ^ = hkng so), theo bien y (xem x l a h a n g so) c i i a he t a dUdc t D (I5a2 - 3a + 15)^ = ( - a ^ + 7a - 8a + 5) (45 - 18a) <^153a'* + 216a^ + 360a = <^a ( I a ^ + 216a2 + 360) = <^ a = a = -2 K h i a = t h i D = 5, Z)^ = 15, s u y r a x = ^ = ^ y K h i a = - t h i D = , D^, = 81, suy T& x = He d a cho c6 h a i n g h i e m | ^ = Q va | ^ ^ y = • Bai t o a n 70 Giai he phUdng ,, (1) (2) Hwdng d i n Lan htdt dao ham hai vc phitdng t r i n h thi't n h a t theo bicn x I2 + ^"o ~ f ta se thu ^dt u = x + 2, Khi x^ + x + + x y - x + 22/ - + 2y2 - 12?/ + 18 = 3x2 + 12x + 12 - x y + x - 22/ + - y2 + 62/ - = 268 i -2 V i d u Tic mot he dang cap bac hai, hang each tinh tien nghiem, V = y - trinh x + 3y2 + 4xy - 18x - 22y + 31 = \ x + y + x y + x - y + = (xeni y l a hang so), theo bien y (xem x la hang so) ciia h e t a dUdc phudng p h a p d a t r i n h b a y t r e n t a luon giai dUdc c a c he b a c h a i tong quat " -2 Tit C O phep dat x = u - va y = ?; + Lufu y V i c a c p h u d n g t r i n h d a thi'rc b a c khong q u a luon giai diWc n c n vdi dUdc mdt he bac hai tSng qudt Xet he I r2x + y + l = ^ r x = \y + X - 10 = ^ \ = „ r 2x + 4y - 18 = ^ / x = \x + y - 2 = ^ \ = -5 -3 • v a y , thitc hien phep doi bien x = - + iz, y = + t;, t a ditdc Jl ju"^+ 3v^+ 4uv = 2u2 + 4?;2 + 2?iu = (3) (4) ' .f ' MAO '' ' (11) He la he dang cap, c6 the giai theo each thong thucJng, nhung I m i y la tr(t hai phUdng t r i n h (3) va (4) v^ theo ve t a c6 u'^+ v'^-2uv = ^ u = y \= V = 2\/2 -1 Tacohe ( / ) o { w ! = l p a p s He CO nghiem l ^ - 2^) = + 1- ^ ^ , , ^ ^ , , f x2 - 2xy + 2y + 15 = B a i t o a n 74 Giai he phuang tnnh | _ ^^^^ + ^2 ^ ^ Vay ( / ) CO nghiem D a p so He CO n g h i § m (x; y) la {2^2 + 1; ^ + l ) ; ( - \ / + i ; - ^ + 1) 2v/2' 5; —i + 4,4 B a i t o a n 71 ( D l n g h i O l y m p i c / / 1 ) Giai he phuang trinh Phifdng phap dung t i n h ddn dieu c u a h a m so T i n h ddn dion c.iia ham so la mot cong cu hfru hic\ dfi sang tac va giai phUdng t r i n h , van de da diTdc t r i n h bay ci bai 1.3 : PhUdng phap dua phUdng t r i n h ve phirdng t r i n h ham (5 trang 15) Trong bai t a se khai thac t i n h (Idu dicu ham so dc giai ho phUdng t r i n h M o t so van dc ve pluidng phap giai da CO a t r a n g 15 nen khong neu r a d day m a t a se di vao nhiing vi du, bai toan cv the a.2 + y _ ^ + y + l l = x2 + 4y2 - 2a;y - X + 4y - 12 = H i f d n g d a n N c u tinh tao nhhi nhan thi thay ugay rang day la bai toan dS ' 3.7; — 23 ' lay hai phitdng trinh trijf nliau t a ditdc y = —, the vac phitdng trinh t o a n 75 Gidi he / W I T ^ + ^1+7) = (1) B a i t o a n Giai h^ ^ ^^^^ - 2xy + = 4xy + 6x + (2) thii nhat ciia he dUdc phUdng trinh bac va may man la phiTdng trinh CO t i hai nghiem "dep" x = va x = B a i t o a n Giai he phudng trinh ( H\+ f 2^," ' ^ \x^ + y^ + x + y-4 G i a i Dieu kien 6x - 2xy + > T a c6 + 2/+ = (1) = (2) phudng trinh b^c h a i vdi an y, x la thara so, c6 = x + va -1 P = - x + 5a; _ ^ - (x - 2) (2x - 1) = {-x + 2) (2x - ) ( ^ x + 6x + 13^ > 0, do fit) dong bien tren R nen - I) = x\ x/2x2 + x - = 3x 25x2 (3) \/2x2 + 6x 4-1 = - x (4) - ; - "T • (x + y - 2) (2x - y - 0) = [ ^ c ( ) ^ { ^ i x + l = 9x2 = [Tac6(4) {2<t6x.l=4x2 ^ ^= thu dUdc y = - X , y = 2x - bang each tach, them bdt, ro rang kho khS" hdn nhieu so vdi phan tich - x ^ 53 _ (2 - x ) (2x - 1) | | y he ( / ) CO hai nghiem (x; y) - (1; - ) ; (x; y) = ( B a i t o a n Giai h$ phuang trinh ^ • x ' -re n - ' i ' B am toan 76 Giai he - - ^ + 2y = - U r - x y + 2x = - 270 ^/W^ < ^ x \ / x + 6x + = 2x2 + 6x 4-1 - g^2 -i=-0 _ 5^ + y + = ^\^^^ / vdi f{t) = t + x V x + 2x2 + = - x + 6x + ( y = 2-x { y= 2x-l l x + / + x + y - = ' \ + i/ +x + y L i f u y De bien doi 2x2 + x y - - / + \/y2 + y + \/y2 + (1) ^ X = - y T h e vao (2), t a dudc CO hai nghiem yi = - x, y2 — 2x - D a n den He CO hai nghiem (x; ?/) = ( ; ) , (x; ?/) = /n Ma (1) Ition CO nghia vdi m o i x G R, y G R va (1) <^ fix) = fi-y) Ta chon P = {-x + 2) (2x - 1) de ( - x + 2) + (2x - 1) - x + - S Vay (1) (1) ^ X + v/x2 + l = - y + V'y2 + ^ fix) = fi-y), H i f d n g d a n T a c6 (1) o y'^ - ( x + l)y - 2x2 + 5^,- - = T a coi day la ( 1' ^1 + l ) • + l)^( I = zl^^ ^ ; 7 ^ - ( x - ) = y y ^ + 3y + y) (2x - y) + = - x - 3y 271 )• (1) (2) (i) G i a i D i c u kieii x > o > T a c6: ^ x ( x ^ + x^ - 3x + 3) = ^ [ / (2) <^ (x + y){2x - ?;) + + 4(x + jj) + (2x - ?y) = ^{x + y + l){2x-y + 4) = x'* + x2 - 3x + = T^ + (^x - - J l i e m d u y nhat {x;y) = (0,1) ^\/ = 2x + (do t i r dieu kien suy r a x + y + l > - > ) Thay vao (1), t a dUdc : ' B a i t o a n 78 G.dz/.e V \ / ^ ^ + 2x - = v / ^ + 3\ ^ <:»2 (3x - 1) + v / x - = (2x + 3) + V2x + (3) _ 3^ ^ 3 + ^ > nen (4) vo nghiem Vay ( / ) c6 f (41/2 + 1) + (x2 + 1) =6 | ,2^ (2 + y v T l ) = x + 22/(1 (3) 4^ / ( V s T ^ ) - / (y2irr3) / (v^3x^) = / ^ VSx^ = V2F+3 ^ X = =^ y = 12 (2) (^) G i a i Dieu kien : x > Neu x = t h i t i t (1), t a c6 = (sai') Vay gia sii x> 0, chia ca hai ve ciia (2) cho x^, t a dUdc Xet ham so / (i) = 2f + t vdi ( > K h i M a /'(O = 4t + > Vf > nen ham so / dong bicn t r c n [0; +oo), do : + ^/VTT) = i ('i + yjTi^ (3) Xet ham so f (t) = t (l + V T + F ) , vdi t G M, phitdng t r i n h (3) viet lai / ( y ) = / ( i ) T a c6 f'{t) = + ^ T T ? + ^ i = = > 0, V i G , do t i l Vay he phirong t r i n h ( / ) c6 nghiem d u y nhat (x, y) = (4,12) / ( y ) = / ( - ) t a CO 2y = - Thay vao (1) t a dUdc B a i t o a n 7 Gidi he phMng trinh j^s + ^s^^.^'l^o ^ ^ X (2) .x^ + Y tifdng R a t ti.r nhien t a n h i n vao tiifng phiWng t r i n h dc danh gia vdi muc dich t i m m o i quan he gifra hai bien Txi (1) t a thay rang ve l a da thiic doc lap ciia bien x, y va ciing bac N h u vay viec ap dung phitdng phap sut dung t i n h ddn dicu c6 cd h o i cong rat cao V a day cQng l a liic chung t a dung t d i k y t h u a t he so bat d i n h D a u t i c n , t a chon m o t d a thi'tc bat k i lam chuan d (1) De thay nen chon da thiic ben ve t r a i v i n h i n no ddn gian hdn Vdi y tulcing t a dUdc ham so dac trUng f{t) = t^ + t-2, nhuT vay viec ciia chiing t a can lam l a phan tich : y^ + 3y^+Ay = g\y) + (3) Xet ham so f{t) = + i - 2, i G R T a c6 f'{t) = 3^^ + > 0, V( G R Suy / dong bien t r e n E Vay (3) / ( x ) = f{y + 1) <^ x = y + T h e vao (2) : x^ + x^ - 3x2 + 3x = 272 + (x2 + 1) - = 4=> x^ + X - = - (x^ + l ) V ^ (4) cd g'{x) = 3x2 + > va /^/(^•^ = _ f x v ^ + < 0, Vx > V 2v/x J /ay g{x),h{x) ddn dicu ngUdc chieu t r c n (0;+00) va g{l) - h{\) nen (4) cd nghiem d u y nhat x = 1, suy r a y = ^ D o ( / ) cd nghiem (x, y) = ^ , ^ B a i t o a n Giai he phudng r (\/^2^-3x2y R6 rang g{y) c6 dang g{y) = y + h t i t day t a khai trien va dong nhat he so dUdc = N h u vay t a c6 phitdng t r i n h + x - = (y + 1)^ + (y + 1) - 2T6i day t h i y titcfng giai b a i toan da ditdc hoan thien G i a i T i t (1) t a c6 x^ + (x - 1)^ + = X p t cac ham so (x) = x^ + x - 6, / i (x) = - (x^ + l ) ^ x , vdi x G (0; +00) g{y)-2 x3 + x - = ( y + l ) ' + (y + l ) - X x2y - X + trinh + 2) ( v / V T T + l ) = x , / (1) (2) = G i a i Vdi x = hoSc y = t h i thay vao he (I) dan t d i v6 l i G i a sii x 2/ 7^ Phitdng t r i n h (1) tUdng dildng vdi ^^^^tizl^'^ + ^4y2 = 8x2y3 ^ ^\/x2 ^^^-^=^'y + l - 4x2y + X = 2x2y v/4y2 + - 2x2y <!=>\/x2 + + X = 2x2y (V4y2 + + 1) 273 + ^ ^ 2x2y (I) va X -o + + + 1+ J{2yf = 2y ) Xet ham so f{t) = t (^fi?T\ l ) c6 /'(O = + rng nhat he so t a dUdc (3) >0 neri / dong bien tren R T i r (3) t a c6 / ( - I = / (2y) <^ - = 2y <^ 2x1^ = Thf> X vao (2), t a c6 : 2x'^y - x + = < ^ x - x + = < ^ x = 4=>?/ = ^ Ket luan : He c6 nghiem d u y nhat (x; y) = 4; - V 8/ X^ - B a i t o a n 80 Giai he 3X2 + 2= Ta (/) Y t i f d n g Chung t a lai bat dau t i m t o i t i i cai ddn gian t d i phiic tap Tvi (]) de y rang t a da c6 dang g{x) - h{y) nhit mong muon, n h u vay y tu6ng dting t i n h ddn dieu de xet ham dac trUng da xuat hien Se t o t hdn neu g{x), h{y) la ham da thiic Vay t a thijf b i n h phudng de loai bo can thirc : ^ - 3x2 + = + N h u vay t a se dat a = \/y~+3 => y = - 3, y-^y + = {a^ - 3)o = Ham dac t r u n g se la f{t) = - "it Do can phan tich x^-3x2 + = 5^(x)-3g(x) De thay r^ng g{x) = x + T i r ^x^ _ 3(^ ^ ^) - 3x2 + = x^ + 36x2 ^ ^3^2 _ 3^^ ^ ^3 _ 3^ - i la ham dong bien tren [1; + 0 ) N h u vay y x>2 ( x>2 y2 + y > [ yG(-oo;-8]U[0;+cx)) f ^ o (1) ^ x^ - 3x2 + = ^ (3 _ 1)3 _ 3(^ _ 1) {^/yT^f - 3\fyT^- Ta CO s/y + ^ > ^3 > 1, x - > Xet ham dac t n m g f{t) = - 3t, Vi > CO / ' ( < ) = 3*2 - > 0, V < > Suy ham so / dong bien tren [1; + 0 ) , do tit /(x - 1) = /(VyT^) t a CO X - = sfyT^ ^y = x^ -2x-2 T h e vao (2) ta dudc - 3« x = <^(x - 3)(x3 - x2 + 5x - 2) = <^ x^ - x2 + 5x - = Xet Q(x) = x'^ - x2 + 5x - c6 Q'(x) = 3x2 - 2x + > 0, Vx € g^y jay la mot ham dong bicn tren R Lai c6 x > Q(x) > Q(2) = 13 > 0, suy phUdng t r i n h Q{x) = vo nghiem Vay he phildng t r i n h (/) c6 nghiem nhat ( x ; y ) = ( ; l ) N h a n x e t NhUng bai toan tren da cho thay c6 nhieu each de dUa hai vi cua mot phuang trinh ve ham dac triing Tuy nhien mot so bai loan kho hdn sc dai hoi phai bicn doi cdc phuang trinh cua he de' tim ham dac trUng B a i t o a n 81 Giai he phudng trinh | + 3y) - I x(y* - 2) = j G i a i Vdi x = 0, the vao (I) thay v6 H G i a sii x 7^ 0, tit (I) ta c6 = — , vdi fit) - X 275 274 ^ ^ CO + 3y = x^ - 3x2 + = (x + - 3\/^T3 ^x"^ - 4x3 + 8x2 - 17x + = Cong viec tiep theo la t i m ham dac tritng De thay h{y) = + 3y2 la lira chon t o t v i day la ham so ddn gian va dong bicn tren [0; + 0 ) Ta se c6 g d n g phan tich {x^ - 3x2 ^ 2f = q^{x) + 372(x) Dong nhat he so se t i m diTcic q{x) = x2 - 2x - Suy x2 - 2x - = y (chu y dieu kien c6 nghiem l a x^ - 3x2 + = (x - i ) ( ^ _ 2x - 2) > x2 - 2x - > 0, X > 2) Nhmig cau hoi dat la, vice k h a i tridn va dong nhat he so v6i (.r^ - 3x2 _|_ 2)2 j^jj/^ phiic tap Lai chii y rang ham so dac trUng khong phai la d u y nhat Lieu co mot ham so nao ddn gian hdn ? Vay dieu tU nhien la t a se d i t i m each d a t an p h u : mot ham chiia can nao dg khong phai luy thita De y rang ^ Vt.f 9(x - 2) - y2 ^ 8y <^ 9(x - 2) = (x2 - 2x - 2f + 8(x2 - 2x - 2) (x^ - x + f = y3 +32/2 (1) <^ x^ - 3x2 + = ham dac trUng /(() = tiffing da ro rang G i a i Dieu kien I ^ (2) (1) ^ = - Do (x - 1)^ - 3(x - 1) = i^/y + zf + 1+ , \ / - = 36 = 36^ - = 63 - 36 = t^ + St Co fit) f 2x2_^x'^ + 4x - = 2x^(2 - y)^^ \VFT2= -^14 - X y / S ^ ^ + ( ^x + y + l + ^/¥Ty = \ + xy + + >/?y2 + xy + = = 3*2 + > 0,Vt e K, suy / la ham dong bien tren M, (j^ y = — T h a y vao phUdng trinh thu: nhat cua he, t a dUdc X x ^ ( + - ) = « - x ^ + 3x2 - = < ^ x X TM ' -, x = ^ -l a) -y* -2 = - 2y Zx-Zy + yr=^ - Z^2y - y2 + = lai ta thay he ( I ) c6 hai nghiem (x; y) = ( - ; - ) , (x; y) = ( ^ ; 2) L i f u y Bang each dat t = -,t?i 4.5 dua vc ho doi xilfng loai hai theo t va y 4.5.1 B a i t o a n 82 Gidi he phUcJng trinh x2 +l (2) G i a i D i l u kien ( | \ " * o l ^ n ° ^ ^ t ham s6 : f{t) = e\t +l),te \^x + y + Z > yj [0, + 00) '0 V i /'(/,) = e^{t + 1) + e* > 0, V^, > nen / la ham dong bien tren [0; +oo) (1) <^ ế(x2 + 1) = ey\y^ + \ ) ^ / ( x ^ ) = f{y^) ^x^ = y^^x=±y log2 [ ( x + 2y + 6)^] = logs [2(x + y + 2f (2) ^ (x + 2y + 6)^ = 2(x + y + (3) 2f • Neu x — y t h i thay vao (3), t a ditdc (4) (3x + f = 2(2x + 2)2 Theo dieu kien t a c6 x > - L a i c6 (3x + 6)^ - 2(2x + 4)^ = (x + 2)^(27x + 46) > (3x + 6)^ > 2(2x + f D o (3x + 6)^ > 2(2x + 4)^ > 2(2x + ) ^ suy (4) v6 nghiem • Neu X = - y , thay vao (3), t a dUdc : ( - X + f = 2(2)2 < - > ( - x ) ^ = ^ - x = < ^ x = 4=J>j/ = - Vay he (*) da cho c6 nghiem nhat la (x, y) = (4, - ) B a i t o a n 83 Gidi cdc h$ phUdng trinh sau : ?(8x-3)\/2^^-y-4y3 = o I 4x2 _ 8x + 2y^ + y2 _ 2y + = 3/ rx-*(3-; + 55:) = 64 I xy{y'^ + 3y + 3) = 12 + 51X 276 N h a n d a n g v a phu'dng p h a p g i a i N h a n d a n g (1) 31og2(x + 22/ + 6) = 21og2(x + y + 2) + l H e lap ba an (hoan vi vong quanh) He lap ba an la he phudng t r i n h c6 dang I = /(y) y — f(z) X z = (trong / la ham so) f(x) {too:; (*) Phu'dng p h a p giai Xet he lap ba an (*), vdi / la ham so CO tap xac dinh la D, tap gia trj la T va T C D , ham so / dong bien tron T C a c h 1: Doan nghiem roi chiing minh he c6 nghiem d u y nhat T h u d n g de cluing minh he c6 nghiem nhat ta cong ba phudng trinh ciia he ve theo vc, sau suy x = y = z C a c h 2: Tir T C D suy / ( x ) , / ( / ( x ) ) va / ( / ( / ( x ) ) ) thuoc D D i (x; y; z) la nghiem ciia he thi x e T Neu x > / ( x ) thi / tang tren T nen ta c6 / ( x ) > / ( / ( x ) ) Vay / ( / ( x ) ) > / ( / ( / ( x ) ) ) Do ^ > / ( x ) > / ( / ( x ) ) > / ( / ( / ( x ) ) ) = x Dieu mau thuan chiing to khong the co x > / ( x ) T u d n g tir cung khong thg CO X < / ( x ) Do / ( x ) = x Viec giai he (*) dUdc quy ve giai phUdng trinh / ( x ) = x H d n nfra ta c6 : X = fiy) y = z= Hz X = fiy M y = ( fix z = X = f(y) I z = /(/(/(z))) fifiy)) 'J I z = I{z z = f{z) I I Jhii y He lap ba an ducic gidi hhng each dm he vi dang cd ban {flz).C = f(x) if f 277 vdi A, B,C >\ f{y), f{z) > {xem bdi todn 101 d trang 286) hoQ.c diia ve he ( fHx) = {y-x)>'.A f\y) = {z-yr.B I , \= tr y > va tir (2) ta c6 ^3 ; ^; 2> _ = %y{y - 2) > i : * Vay = (x - 2f + (y - 2)^ + (2 - 2)^ > Day la dieu vo Ii • Neu < X < (ta CO x > vi theo (3) thi x^ = 6(2 - 1)^ + > 0) thi tif (3) suy 62(2 - ) = X ^ - < = ^ < < {x-zr.C, A > 0, B > 0, C > vd k, m, n la cdc so nguyen duang li {xem bdi todn 101 d trang 286) C h u y Khi ham f khong thod cdc dieu kien da noi phan phuang phdp gidi thi ta phdi c6 nhUng each xii li khdc, chang han xem bdi todn 90 d trang 280, bdi todn 109 d trang 292, bdi todn 99 d trang 284, bdi todn 107 d trang 289 Ket hop vdi (2) suy < y < Vay = (x - 2)^ + (y - 2)^ + ( - 2)^ < Day la dieu vo li Vay x = 2, tir (1) ta c6 y = 2, thay y = vao (2) ta c6 ^ = Vay (2; 2; 2) la nghiem nhat ciia he 4.5.2 C h u y Doi vdi he lap ba an thi c6 mot sai lam rat tinh vi, kho phdt hien la sai lam: Do x, y, cd vai trb nhu nen khong mat tinh tSng qudt gid sti X > y > z " Thuc x, y, hodn vi vbng quanh nen phdi xet hai thii tU khdc x > y > z vd y > x > z C a c bai toan C Bai toan 84 Gidi he phMng trinh I I _ + 12x - = (1) - 6y^ + I2y - = (2) - 62^ + 12z - = (3) y3 Bai toan 85 Gidi cdc he phudng trinh sau Giai X = f{z] C a c h He viet lai ^ j / = / ( x ) vdi / ( x ) = v^6x2 _ a) + Khi ham so • U = /(?/), / xac dinh va hen tuc tren R Tiep theo ta t i m tap gia t r i T cua / Ta c6 ^ fix) / b) d) x^y^+yl+y - \ = zl + zl+z - I z = x-^ +x^ + x- { 12.T2-48X + 64 = ,/ { 12y2-48y + 64 = 2^ I 2 - + 64 = x3 ( x - sin y = _ n ^ ^ - i , ^ ^ • 2x+l = y^ + y^+y 2y + l = z^ + z'^ + z 22 + = x^ + x^ + X x^ - 9y2 + 27y - 27 = y3 - 92^ + 272 - 27 = 2^ - x + 27x - 27 = ^ ( x - i x + 8)2 y - sin = - sin X = Vay tap gia t r i cua ham / ( x ) la T - Hufdng dan Xet ham so / ( x ) = sinx c6 tap xac dinh la R va tap gia t r i la [ - ; 1], / dong bien tren [ - ; 1] He da cho viet lai Ta CO (4) ^ x^ - 6x2 + 12x - = <^ (x - 2)^ = ^ X = Vay he da clio Ta chiing minh ditdc x = / ( x ) va | y = [ v ^ ; +oo Ta CO /• dong bien tren -oc [ l ; + o c nen / dong bien tren [ ^ ; + o o ) f(x) Theo phan phitdng phap giai, he da cho f(x) dUdc Viet lai { ^ _ 1^6x2 - 12x + (4) viet hii I ^ f ^ ^ I y= g'{x) = - cos X C a c h 2: Cong ba phiTdng trinh ciia he ve theo ve ta dUtJc (4) Ta CO (2; 2; 2) la mpt nghiem ciia he Ta se chiing minh (2; 2; 2) la nghiem nhat ciia he • Neu X > thi tir (1) ta c6 2/^ - = 6x(x - 2) > 278 y > | ^ Z ^f(^y Xet phvtdng trinh x = sinx tren [ - ; 1] Xet ham so ^(x) = x - sinx tren [ - ; 1] Ta c6: He c6 nghiem nhat la (2; 2; 2) (x - 2)^ + (y - if + (z - 2)^ = ^fi^ > 0, Vx e [ - ; 1] Ham g dong bien tren [ - ; 1], 5(0) = Do x = la nghiem nhat ciia phudng trinh x = sinx tren [ - ; 1] Vay (0;0; 0) la nghiem nhat ciia he nrf Bai toan 87 Giai he phUdng trinh x^ - 3.T2 + x - + ln(x2 - x + 3) = y y3 - 3y2 + 6y - + \n{y'^ -2,y + i) = z 2^ - 32^ + 62 - + ln(22 - 32 + 3) = X 279 G i a i X e t ham so /(.x) = - Sx^ + 6x - + hi(x2 - 3x + 3) H a m so c6 tap xac dinh la M va Vay ham s6 / ( x ) dong bien tren cac khoang , \I \ -cx); - v/3;'VV3'y3;'Vv/3 Hdn niJa t a c6 l i m fix) = ± o o va va lim Vay f{x) dong bien t r e n E He da cho viet lai: < y = / ( x ) x = / W = / ( x ) - X =^ li{x) = / ' ( x ) - > 0,Vx G R Vay li{x) dong bion tren R Hdn nifa /i(2) = Do /i(x) = (*) x = Do I ^ ^ ^ ^ ^ He da cho c6 mot nghiem d u y n h i t l a (2; 2; 2) B a i t o a n 88 Gidi he phiiOng + 32 - + ln(z2 - z + 1) = X 2004 = trmh 30y r \ '2y 30x 1002 ± 2v/550971 r2 +00, hm = -oo va a 7^ ± - K h i d o ! - 3x _ t a n ^ g - tan a 3x2-1 = tan3Q X = t a n a = f{z) = /(tanOa) = tan 27a sau + Az a = ^^^^^ vk cac hoan v i ciia no + 4x 3tan2a-l ~ Vay t a c6 'lie la a ^ 27a - kn + 4y 2004 = 26 (A; e Z ) Vay nghiem ciia h$ la (0;0;0), hoan v i , ( t a n ( ^ ) ; t a n ( ^ ) ; t a n ( ^ ) ' 26 26 / B a i t o a n ( H S G q u o c g i a n a m h o c 0 - 0 , b a n g A ) Gidi h$ .wP,v: B a i t o a n 90 ( D e n g h i t h i O l y m p i c - ) Gidi he phuang ^ — 71 _ I y = tan3a lo <^ z = t a n 9a ' x = : t a n 27a B a i t o a n ( d e nghj t h i O l y m p i c - ) Gidi he phudng D a p so: x = y = z = a e y 2004 = lim X = ±—r^ khong thoa m a n phudng t r i n h nen de x l a nghiem ciia phUdng v3 x^ - 3x trinh t h i x khac ±—7=, k h i y = - — — - Do t a dat x = t a n a , vdi v3 ,„ 3x^-1 trinh ( x^ + 3x - + ln(x^ - X + 1) = y <^ y + y - + l n ( y - y + l ) = I = -oo, Vay tap gia t r i ciia / ( x ) l a R Tkp xac d i n h ciia h a m so / l a con, thitc su cua tap gia t r i ciia ham so / nen t a khong the ap d u n g each giai n h u d a t r i n h bay phan phiTdng phap giai X e t phudng t r i n h x^^ - 3x = y(3x^ - 1) V i T i e p theo t a giai phifcing t r i n h f{x) = x <^ / ( x ) - x = D a t h{x) lim T u o n g t i t nhit cac v i du trutdc t a dUdc: = +0O, r x^ - 3x = y(3x2 - 1) <^ y ^ - 3y = z(3y-^ - 1) G i a i De thay he da cho tiTdng dudng vdi r y - trinh f \/x2-2x + 61og,(6-y)^x - y + 6Iog.,(6-z) = y v / z - 2 + o g ( - x ) = / »iai Dg (x; y; z) l a nghiem ciia he da cho t h i dieu kien l a x, y, z nho hdn p da cho tUdng dirdng vcli •" • /(^O fiz) t r o n g logs (6 y ) log3(6 l o g ( x) Vx^ - 2x + y \/y' - 2y + ' Vz^ - 2 + 280 281 (1) (2) (3) hay Ta ,, , m (2) vdi fix) (3) ( \ogM -y) = fix) log3(6 ^ z] = f{y} \olti6 - x) = f(z) CO fix) = ^ Vx^ -2x B a i t o a n 95 la han, B a i t o a n 96 quat gia siif x = max(.r, y, z) t h i c6 trirdng hop: B a i t o a n 97 N' M Do z Do fix) x>y> tang nen fix) > f{y) > f{z), log3(6 - z) > log3(6 - x) log3(6 -y)> suy ^ij, giani nen suy r x2(x+l) = 2(y3-x) + l trinh I y'^{y + 1) = ( ^ - y) + Giai he phuang I z'^iz + l) = 2{x^-z) /(^) f fix) = gi^y) (1) y3 + y2 ^ 2y = 2z^ + hay <^ / y) - g z) (2 vdi ? + 22 + z - x + I f\z)=9{x) (3) + ^2 + 2( Do dang cln'mg m i n h d i w c / va g la nhOng G i a sii rang (x; y; 2) la nghiem ciia h$ va khong giam > y K l i i t i t (1) va ( ) suy r a ^ 9{y) > 9{z) > f{y) V$.y X > y > z > X y > > 2''° =^ z > x ( x = y = = cos u (n 377 57r1 UG | y , y , y | B a i t o a n 93 ( H S G q u o c g i a n a m h o c 2005-2006, b a n g B ) Gidi he r x3 + 3x2 + 2x - - y < yJ + 3y2 + y - = [ 2^ + 32^ + 22 - = x B a i t o a n 94 Gidi he phuang Hu'dng d a n D i o i i kion (4x2 ^ Gidi he phuang trinh r log5X=:log3(4+y^) < logg y = log3(4 + y/z) I log5^ = log3(4 + \ / i ) ; 282 _ ^ COS x = log2 (8 COS - cos x - ) cosy= log2f8cosx-cos2y-5") cos2= l o g ( c o s y - c o s 2 - ) trinh C h u n g ) Gidi he phuang i (4x2 + l ) x + ( y - ) ^ / ^ \2 + y2 + v t ^ = trinh = (1) ^ ^ j.^ (2) ^^-y^'^^ Phudng t r i n h (1) viet lai ^ (3 ^ y ) ^ - 2y Xet h a m so dac t r i m g : f{t) ^ 2x _ (2\xf_ v/(5-2y)' ^ / ^ = 3" 3*2 ^° ^'^^'^ = — + - > 0,Vt G R Suy / la h a m dong bien t r e n R, t ^ 2x = s/b - 2y T l i a y vao (2) t a ditOc - 2y + y ^ 2^/^^^ = <^ (y - 1)^ + v / ^ ^ = Dat D - y - K h i | f + 2v^_:_4x = L^^J \ =^ \J6 ~ 2v " • X = y = Do he da cho tuong duong v d i ' una kob + l r x3 + x2 + 2x = 2y3 + H i f d n g d a n T a c6 I I <7(f) = 2^3 + va f{t) = ham dong bien t r e n M t6ng quat, c6 the coi x Gidi he phuong B a i t o a n 98 ( D H - A - P h a n = y = Tru'dng h d p 2: x > z > y T i l d n g t i t nhir t r e n suy r a x = y = z Phuong t r i n h g{x) = f{x) c6 nghiem nhat x = Vay h f c6 nghiem nhat la (x;y;z)-(3:3;3) B a i t o a n 92 trinh e^ y — y ev - e!/-^ = z z — _ C „z—xX < 6-y<6-2<6-a;<s^a;<2<y=i>a; 2cosx(cos^y + 1) = (1 + c o s y ) 2 c o s y ( l + cos^z) = (1 + 0 ) ^ cos 2(1 +cos^x) = (1 + c o s x ) ^ T - 2.r + 6) Vx2 - 2.T 4- tang, g(x) = log3(6 - x) la ham giiim vdi x < Neu ix;y;z) la mot nghiem ciia he p h i M n g t r i n h t a chiing m i n h x = y = z K h o n g mat t i n h t6ng (.T2 TrvfSng hdpl trinh + > 0,Vx < 6, suy fix) = Gidi he phuang u = 2x K h i i>2 + V - 2u = 1/ = v/3 - 2v , { v'^ + 2w = Z Dat w = v / ^ ^ K h i { u;2 ^ 2u = [ u2 + 2i; = Neu u < thi w'^=3-2u>l=^w>l=^v'^ =^ti < = - 2u > = =^ u > 3-2w<l 1, ;Den day t a gap m a u t h u a n , vay khong the c6 u < TitOng t U , triTdng h0p u > cung khong t h i xay Vay u = Suy u = y = 2, thoa man d i e u kien 283 = i f = Do x = ' Lixtu y Viec t i m 2x = \/5 — 2y bang t i n h ddn dieu cua ham so ditdc tien hanh bang phUdng phap he so bat d i n h n h u sau : Ta se b a t ' d a u phan tich t i t (1) PhUdng t r i n h c6 x, y tach bict ucn k h a nang d i i n g ddn dieu la cao Vay t a bien doi phUdng t r i n h ve dang g{x) = - a (3- 1+ ^ > Vay ham s6 g d6ng bi^n sin = nen he phitong t r i n h c6 nghi^m nhat x = y = z = n g h i cho k y t h i c h o n h o c s i n h gioi c a c C h u y e n k h u vu-c D u y e n H a i v a D n g B ^ n g B i c B o ) " trinh va (3-y)v/5-2y = = B a i t o a n 100 ( D e h{y)<=>{Ax^+l)x={3-y)./5^ Dat s/5 - 2y = a => y = K h i g'{x) ax^ + 1= + R6 rang p(x) c6 dang mx + n, d u n g he so bat d i n h t a t h u diWc m = 2; n = =^ p{x) = 2x n g h i cho k y t h i c h o n h o c s i n h gioi c a c Gidi x = cos trvfdng —y ,3^3 a?y + - = (2) az + - = X (3) • X X > TT z = cos G i a i X e t h a m so / (x) = cos / (x) = .3v/3 ;= S i n 3v/3 X .373^ K h i atif2 Ta <1 fix) I —:=x V3v/3 z} x = y = g (x) = X z Tit — cos 284 c6 •K / (x) a ^ i t j - > = x Xet ham f f{^)^9{y) I /(^ - x| < |x - y\> |x - y\ \y - z| = |z - x| =i- > CO !• |2 > t; > ^ va t, ^ t^, t a c6 Vay / (t) la ham dong bien tren [2^; +oo) H a m <? ( = < la ham dong bien tren [2v^; +oo) Hg da cho t r d t h a n h \f'{0\\{y-z)\<\y-z\ Titdng tir t a c6 - y\ \y - ^1 < 2v/^ ; > v ^ Xet / (i) = at + j , i e [2v/a; +oo) V6i t, > 2^, T i t do, theo d i n h l i Lagrange t a c6 \x-y\ \fiy)-f{z)\ la 'hoo bat dfing thftc Cauchy, t a c6 ax + - > v ^ =^ y > 2^/^ T i t d n g t i t y ' cos {x,y, xz h$ phiCOng trinh max C)' 10.1 n.feo,} 11: (1) ax-\ =y C h u y e n k h u vulc D u y e n H a i v a D o n g B a n g B a c B o ) Slit X = phuang G i a i T i t he suy xyz ^ va x,y,z cung dau Ta thay ngu ix;y;z) nghi$m ciia he t h i ( - x ; -y; -z) cung la nghieni cua he • Xet trifdng hdp x, y, z cung dudng He da cho titdng ditdng vdi he sau Nhir vay ham so dac t n t n g chinh la /(«) = y + ' Gia trtfdng = x + ^ chinh la ham dac t r i t n g ma t a can t i m Vay can phai phan t i c h {4x^ + l ) x = 4x^ + x = B a i t o a n 99 ( D e xy + = yz ay2 az"^ + 1= Ta hy vong r^ng f{t) he ~ sau vdi a e (^\\ - Gidi ma so: ('»lK,f;|) Clia sit X = max {x; y; z} K h i x>2/=>/(x)>/(2/)=^g(2/)>y(.)=>y>2 3\/3 285 ; ! li.'' y > z =^ f (y) > f {z) ^ (z) > ix) x>y>z>x^x=^y 'han cac ve cua b a phUdng t r i n h vdi t a dudc X (vi X > 0,y > 0,z> = z = Vav ho da cho c6 nghiom \ k x z> _1 = y = z= \x, _ = y^ = z- = Lifu (x - 2) (y - 2) (2 - 2) [(x + 1)2 [y + 1)2 + ) + 0) + 2x3 - 4x - = y2 y + y - y - = z2 ,4 ^ _ 42 _ = x2 Htrdng dan He d a cho viet l a i f (.r^ - 4) + 2r' - 4.T = ?/2 _ { ( y _ ) + y - y = z _ <=> [ {z^ - 4) + 2z3 - 4z = x2 - f ( x - ) ( X + 1)2 + ^ / ^ - ) (y+iy^ + l I {z^-2) iz + lf + Do (x + 1)2 + > 1, (y + 1)2 + > 1, 1)2 + > nen t\l (1) suy d l (x; y; z) la nghieni ciia he t h i y'-2 22-2 (y + 1)' + 1^ ( z + l ) ' + l' = = ~ ( x + 1)2 = y ' - (y+ 1)2^0 2 - (2 + 1)2 = x2 x2-2 Nhan cac ve cua ba phUdng t r i n h vdi t a dUdc Lan htdt thay x = 1, y = 1, = vao ho (*) t a dou t h u dudc kot qua (x; y; 2) = (1; 1; 1) Vay he c6 nghiem d u y nhat (x; y; 2) = (1; 1; 1) ( Bai toan 102 Gidi he phuang trinh sau Giai Ho da cho tirdng ditdng vdi r x - x - =y -2 \-3y-2 = - ^ I 2^ - - = - X [ { 286 422 - 92 + = 3x f (2x-3)2 = ; - X I (2y - 3)^ = (2 - y) i (22-3)2 = ( x - ) D o vg t r a i ciia x>z>y>x=i-x = y = z 4x2 _ I2x + = 47/2 - 12y + = 422 - 122 + = ^^^y^^^ Thay vao ho t a dUdc X (x-2)(x+l)2 = y - (y-2)(y+l); =z - ( - ) ( + l)2 = - ( x - ) He da cho tUdng dudng vdi 4x2-llx + 9= y 4y2 - lOy + = 22 ba phudng t r i n h deu khong am nen x^ - 3x = y y j _ 3y = 2^ - 32 = - x= ( x - l ) ( y - l ) ( - l ) [ ( x + l) ( y + l ) (22 + l ) + l Giai 22-2 x€ {-l,-v^,\/2} y e {-l,-v^,v/2} G {-1,-/2,^2} ^ /3 3 \ He CO nghieni d u y nhat ( x ; y ; 2) = ( 2' 2' j " r) ^^^^.^^^ (x-1) (2/2+n=-(2-l) _ 1) ^ ^ _ _ J) (2 _ 1) (^2 ^ 1^ ^ _ _ 1) ( x722 y _- ^2 y + yx_- 1l^=Jl_- ^2 ^ ) 2x2 - x2 + - = - y Tiit day, ket hdp v d i (1), t a dvtdc (x+1)^ + 2x2 + + y = x2 + , >^.xM-i\ He da cho tUdng dudng vdi | x - | > | 2 - | > | y - | > | x - K | x - | = |22-2| = |y2-2| x2-2 xy'' + x + = y'' + y22 + y + X = 22 + ! Giai (.7;2-2)(.T2 + 2.T + 2) = ? / - ( j / - ) ( / + 22/ + 2) = z _ ( 2 _ ) ( 2 + 22 + ) = a - - x2-2|[(x+l)2+l] = |y2_2| |y'2-2| (y+1)2 + = | ^ - | | 2 - | (2 + 1)2 + = x - !(1) (2 + X= y= 2 = L^n hrdt thay x = 2, y = : , = vao he (*) t a deu t h u dUdc ket qua (x; y; 2) = (2; 2; 2) Vay he c6 nghiem d u y nhat (x; y; 2) = (2; 2; 2) -1 r- y Co t h g sijf dung dao h a m de chiing m i n h h a m so / dong bien nhauh gon hdn Bai toan 101 Giai he phuong trinh = 0^ |ii toan 105 Gidi he phuang trinh 287 C x2 + 2y2 = X (1) < y2 + 22x = y (2) I 22 + 2xy = (3) j , G i a i L a y ( ) trtf ( ) , t a ditdc x'^ - y'^ + 2z{y-x) Lay Hoan v i vong quanh (thay (x; y; z) hdi (y; z; x ) ) t a dvfdc = x-y<^{x-y){x + y-2z-l) = (4) = y-z^{y-z){y + z - 2x - 1) = (5) (2) t i i t (3), t a ditdc y -z'i • T r u d n g hdp : x = y = z T h a y vao he ta dildc 3x^ = x •i^ x e |o, ( x(x + 2z-l) \ + z - x - l tir (6) suy (0;0;0), (1;0;0), (0;1;0), (0;0;1), 3'3'~3 /2 ^3' fx I + y^ + z^ = y + z^ + x^ = z + x^ + y^ = (1), (2), (3) 1, dan din ,s j , , , , , vr ^ - ^ " ' i X = y - y^ y = 42 - 2^ (1) (2) = 4x-x2 (3) »s , Cpng ba p h i T d n g t r i n h ta d U d c x^ + y^ + 2^ = (x + y + 2) <^ p2 - 2g = 3p f,x = y - y y = Az-z^ I = x - x2 f (y-Ufy-S) = 3-x - - 3) = - y I (x - l ) ( x - 3) = - I _ i _ n ' " (5) c (7) , - i-: ^ ^ (x - 3)(y - ) ( - 3)[(x - l ) ( y - \){z - 1) + 1] = trinh (8) T r i t n g h d p 1: x = T h e vao (5), (G), (7) ditdc (x; y; z) = (3; 3; 3) Trifdng h d p 2: y = The vao (5), (6), (7) d u d c (x; y; 2) = (3; 3; 3) | T r i t d n g h d p 3: = The vao (5), (G), (7) d U d c {x\y\z) I Tritdng h d p 4: '; " " ' = (3; 3; 3) x = (y = 0, = la t U d n g t u ) The (7) d U d c = 0, the = vao (G) diWc y = Vay (x; y; 2) = (0; 0; 0) la nghiem ciia he da cho = (4) Trirdng h d p 5: x ^ { , } va y ^ { , } va ^ { , } T i f (8) t a c6 (x - l ) ( y - ) ( - 1) + - ^ (x - 1) ( y - y - + 1) + = (5) •^xyz -xy-xz t a dudc X (1 - x ) + / ( / - 1) = y (1 - y2) + ( x (4) Nhau (5), (6), (7) vdi dudc y {1 - y"") + z{z - 1) + x' {x ~ 1) = Thitc hien (4) - z{5), > Do t a n h a m d U d c x = y = = l a nghiem nen se bien doi he da cho t h a n h (1) (2) (3) + z^ {z-l) c6 G i a i D a t p = x + y + z, q = xy + yz + zx, r = xyz G i a i L a n Ivrdt lay (1) t r i t (2) va lay (2) tru: (3), t a dudc x{l-x^)+y{y-l) (7) { Tim cdc so x > 0, y > 0, z > Q thod man h$ phudng { txl f' Q;^;^) , 3' t i i (6) suy y < 1, do tuf (7) c6 < 1, dan den X < Do he k h o n g d6i k h i thay (x; y; z) hdi mot hoan vj ciia no nen theo tren ta B a i t o a n 106 (7) Vay (x; ?/; 2) = ( ; 1; 1) la nghiem nhat ciia he dUdc t a t ca cac nghiem cua he la x + y'^ + 2^ < ( v o H ) = z (mhu thuan^ 3x-3z-0^x 1, X khac t t o g doi, t a suy r a > {x + 2 - 1) = z = x+l = ^ = ^ x + 2z=l X - z = - I 3'~3'3 y x + y' + z^>3iv6li) Neu {x; z) = (0; 1) f X + y - 2z - = { y + 'z-2x-l=0 nen ^1 • T n r d n g hdp : x = y 7^ z TiT (1) va (5), t a c6 ' Xet t r i r d n g h d p ( x ; y ; 2) 7^ ( ; 1; 1) Neu x > t h i gia thiet y > , > + 2x{z-y) TrUdng hdp : x,y,z y ( l - y ) ( l + y + yx) = ( - ) ( l + x + x ) + x-yz + y + z = 0'^r-q+p = (9) E u n g t a n h d m diTdc x = y = = 1a nghiem nen se bien doi he da cho - 1) <=>x (1 - x ) (1 + X + xz) = y (1 - y) (1 + + z y ) ihanh (6) r x = yf4-y} { y = zh-z) { z = x(4 - x) 288 289 • Ct cat: dang thiic Tiep tuc nhan ba phitdng t r i n h vcii va lutu y den trucJng hdp dang xet (Tritdng hdp 5), t a dudc xyz < ^ (.T <^xyz = xyz (4 - x) (4 - y) (4 - z) <=> {x - 4) - 4) [yz - Ay - 4z + 16) = - 4) (z - 4) = 4cos : r - cos — = cos a o a + 27r -1 -1 - Axy ~ Axz + 16x - Ayz + IQy + 16z - 64 = - <»r-4(7+16p-64=-l<4>167J-4(7 > > 4cos'^ — ; + r = 63 (10) Theo (4), (9), (10) t a C O he i-' \ f p ~\q + r = 63 r-q + p = Q <^ ( \bp -Iq I r = q-p = ?3 (11) ' r ;\ • ^ 2 (15p - 63) - -^—h^ o n = cos : cos (a - 27r) o o " = rv + 27r ;!|*>i" liin ', '' < i ;>\ : n = cos JiWW/ ! /•ifioii ' , j i ' Mil,!, a - 27r Nghiem cua he da cho la (0; 0; ) , (3; 3; 3), , ^ 42 ^ <^ : c o s ( a + 27r) cos — - — TT 57r ^ 77r\ , + c o s - ; + c o s — ; + 2cos — \ 9 / - 3p = ^ 3ju2 _ 30p + 126 - 9p = <»3p2 - 39jy + 126 = <^ p2 _ a + 27r — " ™ o - 27r ^ n: - 27r 4cos — r cos yuy phUdng trinh (*) c6 cac nghiem TCr (11) thitc hien phep the ta diTdc V o „ va cac hoan v i , p = D • Trirdng hop 5a: (p; f/; r ) = (6; 9; 3) Theo d i n h H V i - e t dao t a c6 (x; y; z) la nghiem ciia phirong t r i n h - 6?)^ + 9?; - = 2^7 3+ « , 2v/7 -^"'3'3"" cos a + 27r , 2\/7 _ a cos "'3"^ 27r^ va cac hoan v i L i A i y Bai toan c6 m o t each giai khac ngtln gon hon nhieu, Idi giai tren t u y dai dong n h u n g chiia difng m o t so k y t h u a t rat hay Dat w = + 2, t a dUdc 6^ - 36 = Dat b = 2m, t a diTdc Am^ - 3m = ^ Theo bai toan t r a n g 119, t a duoc B a i t o a n 108 (Middle European Mathematical Olympiad Giai he phUcJng trinh (1) ( 2x^ + 1^3zx \3 + = 3x2/ (2) I 22^ + = 3yz j.' = + c o s ^ , t; = + c o s y , t; = + c o s y • Trirdng h(?p 5b: (p; q; r) = (7; 14; 7) Theo d j n h h' V i - e t dao t a c6 (x; y; z) la nghiem ciia phitdng t r i n h ^ Dat i = a + - , ta ditoc - 7/2 + 14i - = 4n^-3n = - ^ Dat = cosa, vdi a G 2t/3 + l = 2/3 + y3 290 = 3x2 = 32/2 = 3z\ _ (3) (4) (5) )au dang thi'rc d (3) xay <^ x = 1, dau dang th>i:c d (4) xay <^ j/ - 1, \u dang thulc d (5) xay = T i t (3), (4), (5),ta c6 (2x3 + l)(2y^ + l ) ( 2 ^ + TTJ K h i 4?i^ - 3n = cos a = cos ( a + 2-K) + i> ^ 2z^ + l = z^ + z^ + l>3</? (*) (*) Giai • TntcJng hdp : x > 0, y > va z > Theo bat dang t h i i c Co-si, t a co 2x3 + l = x^ + x^ + l > v ^ 7 v/7 - - a = - — D a t a = - y - n , t a dUdc (3) 2012) l)>27xV^^ ^dau dang thitc xay k h i va chi k h i x = y = z = l M a t khac t i t he (*) ta c6 = cos ( o - 27r) (2x3 ^ i)(22/3 + 1)(223 + 1) = 27x^y^z'^, do x = y = z = 291 • Trildng hop : Trong ba so x, y, z c6 diing mot am, hai so lai khoiig am Khong mat tinh t5ng quat, gia sii x < 0, y > 0, > Khi ta c6 > 'ixy'^°=^ 27/3 + > ij r • Trirdng hdp : Trong ba so x, y, z c6 dung hai am, so lai khong am Khong mat tinh tSng quat, gia siif x < 0, j/ < 0, ^ > Khi ta c6 2z^ + > > iyz, dieu mau thuan vdi (3) • TrUdng hdp : Ca ba so x, y, z deu am Khong mat tinh tong quat, gia sit X — max {.T, y, z) Ta c6 , ^ + 2x3 > + Tit X = y > 2, ta =^ 3xz > 3i/2 <!=i> X < y X = y CO 2y3 + 1> + =^ 3xy > 3y2 X < X = Nhif vay x = y = z Thay vao he (*) ta (hruc , ]\/iat khac, ta c6 / ' ( t ) = Qi^ _ 4f + > 0, V< e R Suy / (t) la ham so dong bi^n tren R Khi y > =^ / (y) > / (2) 2+ y > x + y > x Vay 2; = y Suy / (x) = / (y), hay y + x = y + 2, hay x = Thay x = y = z vao he phitdng trinh, ta c6 3x3 - 2x2 - x = ph^((jjjg tj-^nh ^ay c6 nghiem 2; = 0, x = l , x = - - Vay he i)hitdng trinh da cho c6 nghiem (x; y; 2) la (0 ; ; ) , ( l ; l ; l ) , ( - l ; ; ) - LuM y- Neu ta klioiig cpng them x, y, vao tifng phitdng trinh de thu ditdc ham so dong bien / (t) = 3t^ -2i^ + t, teR nhit da trinh bay IcJi giai tren ma xet trite tiep ham so g{t) = Zt^ - 2t^ thi se gap rat nhieu kho khan, ham g{t) = 3^3 - 2t^ khong phai la ham dong bien Thiic te d ky t h i chi CO mot t h i sinh giai difdc tron ven bai bang each dita ve he r X = (loai) 2x3 + = 3x2 ^ (x - l)2(2x + 1) = <!:^ X - Vay nghiem ciia he la ix;y;z) 4.5.3 = (1; 1; 1), {x;y;z) = 323 -| z y 4- x = 3x3 - 2x2 + X + y = 3y3 - 2y2 + y X + 2- = (-^;-^; -|) 2' 2' 2^ Phu'dng p h a p sang tac cac bai toan ve he lap b a § n V i d u Xet mot ham so thoa man hai diiu kien : f{t) khong phdi la ham dong bien tren R va f{t) + t la ham dong bien tren R Chang han ham so (x = f(z) - 2*2, t G R Xet he \ = f (x) Ta ducfc bai toan sau [ ^ = fly)B a i t o a n 109 ( H S G G i a L a i , b a n g A , n a m h o c - 1 ) Gidi he phiCdng trinh V i d u Xuat phdt tic (x + l)(y + 1) = z + 1, co xy = - x - y Tii day Ian hidt thay bo (x;y) bdi {y;z), {z;x) ta dUcJc bai toan 110 Luu y rang hay va kho cua de bai phu thuoc rat nhieu vao phuong trinh xuat phdt B a i t o a n 110 ( C a n a d a N a t i o n a l O l y m p i a d 2004) Gidi hephuang xy xz yz f (t) = X = 323 - 2^2 y = 3x3-2x2 (*) l2-3y3-2y2 G i a i Gia siif x = max {x, y, 2}, the thi x > y > hoS-c x> z>y Xet trudng hdp x> y > z (trUdng hdp x > z > y titdng tit va cac nghiem triing vdi cac nghiem cua tnrdng hdp da xet) He phitdng trinh da cho titdng ditdng vcti X + = 323 - 22^ + y + X = 3x3 - 2x2 + X + v = 3y3 - 2y2 + y ( { Xet ham so / {t) = 3*3 -2t^ + t,te R Khi do, he (*) c6 dang r x + z = f(z) < y + x = J (x) [ z + y = f(y)292 • = = = trinh z- X - y y - X - 'z X - yz l a i Cong hai ve ciia phitdng trinh thi't nhat vdi (x + y + 1) ta difdc xy + x + y + l = + l - ^ ( x + l)(y + 1) = + Tudng tit ta CO (7)^ f ( x + l ) ( y + ) = +1 { x + l)(2 + l ) = y + l [ (y + l)(2 + l ) = x + l (1) (2) (3) Nhan (1), (2), (3) theo ve ta dudc [(x + l)(y + 1)(2 + 1)]2 = (x + l)(y + 1)(2 + 1) Neu X ^7^-1 = - thi tit (1), (2), (3) suy y = = - Gia siS x Khi ( ) ^ ( x + l ) ( y + l)(2 + l ) = l 293 7^ (4) -1, y 7^ - va T i f (5) va (3) suy (x 4-1)^ = 1<=> a;^ + 2x = • Thay x = vao he ( / ) t a ditdc x e {0, - } '/ipn n = 3, rfo 3p2 - 30p + 27 = y = z ^ y;z) = i-2;-2) hQnp= • Thay x = - vao he (/) t a dUdc -2y = ^+ p2 - lOp + = x'* + 3x = ^^.^ ^ ^ T-(/: /mm so ytay I , ta dUcJc ham so f{x) ^ y^l] = X mx Tiirdc het (le y rang he so cua nx^ + p nhimg van khong lam mat t i n h long quat Ta c6 = x"^ + , , x^ + mx f{x) - X = K nx'^ + p Lai B a i t o a n 111 Giai he phihlmj X a y dvfng m o t so Idp h a m de s a n g t a c b a i t o a n m d i M o t m o i l i e n h e giiJa h e phvfdng t r i n h v a d a y s6 L d p h a m f{x) x = x^ + mx-nx'^-px {I - n) x^ + {m - p) x nx'^ +p nx^ + p ^^^2 / \^) — jj-j ^ (x^ _ + x i = a; Xn+i = >, <• ^ ^ / _ ^ ^ , V x € R K h i de dang thay rang fix) - X = - x ( x - 2) 3x2 + ; /(x)-x = 0^xG {0,-1,1} (nx2 + pf x = ±1 +pf Dicni kien de nx^ + (3p - nin)x'^ + inp c6 dang [Ax^ + (luc t a c6 A / > t A = (3p - muY - ^mnp = ^ \ - mmnp + m^n^ = I + / ' ( x ) > 0, suy r a ham so / dong bieu) la 294 () + :> ( + _^-\«- X e ( - 0 ; - ) t i l l / ( x ) thuoc (-00;-1) (i - va fix) > X Do v6i m o i n = , , t h i G (-00;-1), hay 14 ^han tren T a co X2 = / ( x i ) = / ( a ) > a = xi, ma fix) 295 -1 T i f bang ben t a c6: Vdi moi x„ • {^'n }t=i 1: a < IVu-dng h d p r + ^ / " > ^ \^ - lOnp + 3n^ = 0 /(.V) V i d u C/ipn m — A"/// (/u n > \2 - 30np + 9n2 = = ^"^^'^ ^ (3x2 + 1)2' nx'^ + {3p - mn) x^ + mp [nx^ r ,Vn = l , , =/(x„),VnGN* Taco ] r ~ ^2 (nx^ + v) " r •/» > 34 + Giai 3nx'* + 'ipx^ + mnx^ + mp - 2nx'' - 2mnx'^ ^ trinh Chung minh rang day so tren cd gidi han hHu han vd tim gidi han Xn+i 77ix) ^ B a i t o a n 112 ( D e t h i H S G quoc g i a , n a m h o c 9 - 9 , b a n g B ) Cho so thuc a Xet day so { a ; „ } ^ ^ nhu sau: C a c h X e t ham so / ( x ) = ^ (3y2 + 1) = y3 ^ 3,^ y (3^2 + l\= + 'Sz z hz^ + 1) = T^ + 3x ' bang CO (3x2 f{z) Wo) Nghierii ciia he la ( ; ; ) , ( - ; - ; - ) , ( - ; - ; ) va cac hoan vi.,) 4.5.4 tinel ki duac fiai hdi toan sau, Id hdi toan 112 vd bdi loan 111 De thn duac bai loan vi he doi xttng ba an, ta chi can xet he 2-y y = yz = p = ;V;:';;,tiJ ; day so dong bi6n 'it gia thiet ta c6 tren ( - 0 ; - ) nen quy nap theo n suy x„+i > x„, Vn = 1,2 Vay khj ^ ^= o < - thi day { x „ } ^ ^ tang va bi chan tren nen hoi tu Dat hm x„ = 61 thoa man dieu kifn 61 < - va 61 G {0, - , } Do 61 = - hay Hin .T„ = -1 ' n—•+00 Xn + di • • 4-1 +3xn-i n Q ^~ ^ n - l + 3x„_i - - x"•^ ^ _,1i +- r3 x „ _ i ~- ^1 _ \(x„_i - 1)3 5~"2—r~; = 3-<-i + ~ 3xl_, + 1— • _ X ^ _ i + 3xf,_j + 3x 34-1 + 34-1 + • 34-1 + vay Tru'5ng h d p : a = - Khi x„ = - l , V n G N*, suy hm x„ = - x„ - 1 /X„_i - Trirdng h d p 3: - < a < Tit bang tren ta c6: V6i moi x G (-1;0) thi i/(x,r"-'"' /(x) G (-1;0) va /(x) < x Do v6i moi n = 1,2, thi x„ G (-1;0), vay day so {xn}n=i chan ditdi Ta c6 X2 = f{xi) = f{a) < a = xi, dong bien tren (-1;0) nen quy nap theo n suy x„+i < Xn,yn ma XI f{x) = 1,2, at b = •ri - xi + Vay - < a < thi day {x„},"^^ giam va bi chan dudi nen hoi tu Dat hm x„ = 62, h2 thoa man dicu kicn 62 = n—>+oo 3,-1 Khi x„-l Xn + hm x„ < x i = a < va n—»+oo 6+1 = x„ - = 6x„ + <^ x„ = l ' • ' "~" " 1-f ay neu a = - thi x„ = - l V / t G N*, neu a 7^ - thi 62 G { , - , } Do 62 = ~ , vay hm x„ = ~ n—>+oo + Trxidns hdp 4: a = Khi T„ = Vn = 1,2, Suy hm T„ = n—»+oo Tru-dng h d p 5: < a < Tit bang tren ta c6: Vdi moi x G (0; 1) thi /(x) G (0; 1) va f{x) > x Do vdi moi n = 1,2, thi x„ G (0; 1), suy day so {x„}^^^ bi chan tren Ta co X = /(xi) = /(a) > a = x i , ma /(x) dong bien tren (0; 1) nen quy nap theo n suy Xn+l > x„,Vn = 1,2, Vay < a < thi day { x „ } ^ ^ tang va bi chan tren nen hoi t u Dat hm x„ = 63, ^3 thoa man dieu kien bs = hm x„ > x i = a > va ri—'+00 63 G {0, - , } Do 63 = hay hm x„ = n—»+oo Xri = 'a- a+1 O M - I ^ , V n = l,2, n—>+oo T r i f d n g h d p : < a Vdi moi x G (1; +00) thi /(x) G (1; + 0 ) va /(x) < x Do vdi moi n = 1,2, thi x„ G (l;+oo), vay day so { x „ } ^ ^ bi chSn difdi Ta c6 X = f{x\) = f{a) < a = xi, ma /(x) dong bien tren ( ; + 0 ) nen quy nap theo n suy x„+i < x„,Vn = 1,2, Vay I < a thi day {xn}n^i giam va bi chan dirdi nen hoi tu Dat hm x„ = 64, 64 thoa n—•+00 ^ - , ta CO cac bien doi titdng ditcJiig sau a- = l < = > a - l = a + l < ^ a = ^v6 nghiem j a- = - l = > a - l = - a - l < ! ^ a = « +1 a - < <^ {a -ly < {a+ ily\ -2a < 2a a> a+1 a > 1'^ {a - if > (n + 1)^ -2a >2n<^a<0 a +1 Cach De thay rang vdi moi n, luon ton tai x„ Xet ham so /(x) = X X + r1 , ' ' ^ ay ta xet cac tritdng hdp sau : irdng h d p : a = Khi x„ = 0, Vn = 1,2, Suy hm x„ = n-»+oo xidng h d p : a = - Khi x„ = - l , V n G N* Suy hm x„ = - n-»+oo ifdng h d p 3: a > Khi mfm dieu kien 64 > va 64 G 10 - , ) Do 64 = hay hm x„ = n—»+oo (*) - ^ Tru-dng h d p 6: a = Khi T„ = 1, Vn = 1, , Do hm x„ =^ hm »i—»+oo o - i y a+iy = 0^' hm x„ = n—'+oo 297 296 + / T r i r d n g hdp 4: - 7^ a < K h i > a+1 V i d u ' Ta Hm fa- ^ — 1\ {*) = 0=^ Hm n—>+oo n—>+oo Xn — ( x = fiy) !• i z = /(x)-.T duac bai toan B a i t o a n 113 -y ( X (3y2 + l) = {y - \ 3x2 + , —1 T a cdn c6 thi lam kho hon bai toan 111 nhu sau Xet ; he ,/) I z = f\x) t o a n 114 ( D e n g h i t h i O l y m p i c - - 0 ) Giai va bien ( 2x(2/2 + a2) = y(j/2 + 9a2) iheo tham s6 a) he phudng trinh < 2^(22 + a2) = z{z'^ + 9ffl2) l 22(x2 + a2) = x ( x + 9a2) X (3y'^ +l)=2{yy^) y (3z^ + = - z^) z (3x2 + = (x - x^) Gidi he phUcJng trinh f ( x + y = f{y) y + z = f{z) \ + X= f{x), if X > y > z ho&c x > z > y X e t trirdng hop x > y > z (trifdng hdp x > z > y titOng t i t va cac nghiem t r i i n g vdi cac nghiem ciia tritdng h d p da x e t ) K h i x>V=>f {^) > f {y) =^ 3; z > X + y z > y V a y y = z S u y / (y) = f {z),hay vao he phiTdng t r i n h , t a c6 x + y ^ y +z, ludn (1) r 2xy2 = 2^3 iai K h i a = 0, he (1) trd t h a n h < 2yz'^ = z^ Sx^y^z^ = x^y^z^ S u y r a I 22x2 ^ ^.3 xyz = V a y {!) <^ x = y = z = T i t day v c sau t a x e t a 7^ X e t h a m so ,ii'>i-j x'' + 3x ^ ^ la ham dong bien G i a s i i x = m a x { x , y , } , the thi nffui ,.';;o!i Chi f, _ x(x2 + 9a2) _ - x ^ + 7a2x 2{x' 2(x2 + a2) X - X — / W \ = = ^ ^ [ x = ±av/7 \/3a hay x = z T h a y x = y = z fhy ham so / dong bien trcn R L a p luan hoan toan tUdng i\i nhit each d (3x2 + 1) = (x - x^) ^ 5x^ - X = ^ X G jo, luc phitdng phap giai, t a dildc aoit u = - x = x = ±V^ft X = (1) ^ { X = /(x)' ^ V a y he phvtdng t r i n h da cho c6 nghiem (x; y; z) la (0;0;0), x ( x 4-9«2) =2 sau = fiy) -y y = f(z)-z z = j{x)-x X m = Qa^, to fto/c /icVm so Tif /la"'- so ta thiet he duoc hai bai toan sau, la bai toan 114 va bai odn 115 De' thu duac bai toan vi he doi xiing ba an, ta chi can xet he Hu-dng d a n H e viet lai vdi / ( x ) = Chon p = m, = < • 2/ = = = y = z = as/l [ X = ij = = - a \ / X = X I^et luan: K h i a = he (1) c6 nghiem d u y nhat l a (0; 0; 0) K h i a 7^ 0, h ^ (1) l o b a nghiem l a ( ; ; ) , ( a v / ; a v / ; a v / ) , ( - a v / ; - a - / ? ; - a v / ) ; ' ' V i d u Chon n = Khi Jai t o a n 115 Cho so thiCc a > Xet day so nhu sau: x„(x2+9a2) Xi 9p2 _ 20mp + 4?u2 = => (—Y \rn/ {xn}n=i - 20 f — ) + = \m/ 298 fChiing I m 9" = b; Xn+i — ,1 • , ''• • 2(x2+a2) minh rhng vdi moi so thuc b, day so tren c6 gidi hg,n hHu han vd tim tgidi han ! 299 2(x^ + a^) '^^ ^ ^ ' ^^'^^ ^ ° G i a i X c t h a m so /(x) = Xn+i dang thay rang = / ( a - „ ) , V n G N * T a c6 2(x2 + o2) 2(x2 + a2) f r2 - Srj2^2 = 2(x2 + a2)2 ^ 0; /'(^) = ^ X = X = X = X = ±a\/7 \/3a -\/3a (jY-u'dng h d p 5: < < a\/7 T i r bang tren t a c6: Vdi m o i x € (0; a\/7) t h i ({x) G (0; a\/7) va /(x) > x Do v d i m o i n = , , t h i x „ G (0;a>/7), jjay day so {xn}n=\i chan tren T a cd X = / ( x i ) = /(a) > a = x i , m a /(x) jQjig bien tren (0; a\/l) nen quy nap theo n suy r a a;n+i > X n , V n = 1,2, ^ ^ay k h i < < a\/7 t h i day { x „ } ^ ^ tang va b i chan tren nen hoi t u D a t = i-s, k h i L thoa man cheu kien lim = j,._,+0C' + Tru*dng h d p : < -a\/7 T i t bang ben t a c6: Vdi nioi X e (-oo; -aV7) t h i /(x) thufx- ( - o o ; - a / ) va /(x) > x Do vdi m o i n = 1,2, thi b i chan tren Ta c6 dong bicn tren (-oo; -as/l) X2 = f{xi) = f{a) > a = x i , m a /(a) - a v ^ , « V } Do L = as/l hay fru'cing h d p 6: = a\/7 K h i x„ = av/7 Do Trifdng h d p 7: > a\/7 T i r bang tren t a co: Vdi m o i x G (aV^; + 0 ) t h i /(x) G ( a v ^ ; + 0 ) va /(x) < x Do v d i n = , , t h i x„ G (a>/7; + 0 ) Ta CO X = / ( x i ) = /(a) < a = x i , m a /(x) dong bicn tren (av/7; +cx)) nen {xn}n=i S'^"' "^^ b i chan dudi nen hoi t u D a t lim Xn — a\/l n-*+oo , Vf d u Chon n = Khi V n—>+oo - 30mj!; + 9m' = =^ = ^ ( ^ y - m + = 0= L i G {O,-a\/7, a\/7} Chpn — = 771 n—+00 m — 3p CTion m = a T r i f d n g h d p 2: = -a\/7 K h i x „ = -a\/7, Vn = , , suy r a lim Trtfdng h d p : -nV7 x„ = p = a, to duac ham so x'^ + a x ,, ^ , -ay/l < x Do x, G (-a\/7;0), vay day so {xn]l^i t o a n 1 Gidi vd bi$n luan {theo tham so a khong am) h$ phuang (3(/2 + a)x = y{y^ + 3a) (32^ + a)y = ^(^2 + 3a) (3x2 + a)z = x(x2 + 3a) < a = x i , m a /(x) dong bicn nen q u y nap theo 7i suy r a x ^ + i < x„,Vn = , , Vay - a \ / l < a < t h i day { x „ } ^ ^ giam va b i chan ditdi nen hoi t u Dat hm m ^il ham so ta thiet ke duoc bdi todn 1 vd bdi todn 1 bi chan dudi T a c6 X = /(a^i) = /(a) tren {-a^/l;Q) , < < TiT bang tren t a c6: Vdi m o i x G (-flV7;0) t h i /(x) G (-av/7;0) va f{x) - lOmp + 3m^ = = A, m L h m x „ = -ov/7 n—>+oo h m x „ = L , k h i L thoa n—^+cx> x „ = L i , k h i Li thoa man dieu kien L\ -a\/7 va Do L i = -a\/7 hay h m x„ = a\/7 Tl—» + 0 Vay k h i < - A V ? t h i day {.T„}^^ taug va b i chau t r e n nen hoi t u Dat hni h m x„ = a\/l man dieu kien L > a\/7 va L G {O, -a\/7, av/7} D o L = aV7, hay nen q u y nap theo n suy r a Xn+i > x „ , V n = 1,2, V ; S h m x „ > x\ a > Q va n—>+oo quy nap theo n suy r a x„+i < x„,Vn = 1,2, Vay k h i b > a\/7 t h i day x „ G ( - o o ; - a \ / ) , hay day so {xn}n=i .•ifv: inlr, " ' f x „ = L , k h i L thoa man dieu kien L = n—>+oo h m x „ < x i = < va n—•+00 L2 G {0 - a v / , a\/7} D o L = - a \ / , vay h m x„ = Trtrdng h d p : /; ^ K h i r„ = 0, Vn = , , Suy r a -a\fl h m T„ = t o a n 1 Cho a > c6 dinh Xet day so { x ^ } nhit sau: ( XQ>0 ""^^= jxl + 3a) 3x2+a ' V U G n—•+c» 300 301 ' trinh Tim tat cd c/ir so dudng X Q cho day hap hay tinh gidi han cua day co yidi han vd trony cdc trudu,^ B a i t o a n Cho trade hai so a, b thoa man dieu kien b > a > Q Gidi he phUdng trinh bai toan 117 nhit sau : X e t ham so g{x) = ( {a + b)x =^ y'^ + ab \ + b)y = zl^ab ,„ {a + b)z = x^ + ab • L u ^ y T a ( o thS t i m du'rtc so hang tfing quat cua day so t r n y hoi clio tioi|„ I T i r gia thiot (-a co / > £)ap so He phirdng t r i n h co hai nghiem (o; a; a), (6; 6; h) \fa = Xn- x f , _ i + 3ax„-i 3xf,_i + a - \fa nx^ xl^ - 3x!^„ v/^ + 3ax„,^i - ay/o _ ( x „ - i - >/») 3x2_, + a 3x2_, + a Titdng ti.r t a c6 x „ -j- Xn = ( x n - i + \/uy 3xti +a — Tntdc het de y rang he so ciia + p nhifng van khong lam mat t i n h tong quat T a co f'( \ (^^'' Bdi vfxv ' bc^ng ° " '^") ("^^ +P) ~ 4nx^ (x^ + m x ) 5nx^ + 5;)x^ + mnx^ + rnp — 4nx^ — 4mnx^ (nx" + pf /x - va \ L d p h a m f { x ) = _ nx^ + {bp - 3mn) x^ + mp !\: •»>l ,,,1 {nx'^+pf Dat T Xn - ^/o y/o r- - x „ ++ 'v/ft Xn ( a + 1) \/a r- Dieu kien dg cho nx* + {bp - 3mn) x^ + mp co dang {Ax"^ + Bf / ' ( x ) > 0, suy r a ham so / dong bien) la (luc r ">« ^ ^/">0 I A = (5p - 3mn)^ - 4mnp = ^ \^ - 34mnp + Qm^n^ = ° 3" ''i d u Chon n = 2, Vay Xn = X() - V" 3" 25p2 - 68;/tp + 36/At'^ - 68 (£•) + 36 = = Q<^2b • 111" B a i t o a n 1 ( D e n g h i O L Y M P I C / / 0 ) Cho day { x „ } , vdi ''C/ipn — =z ^ xo = \/2 a>2 [xj + 3a) - _p = m 18 P_ =^ p = C/jpn / n = 25a'*, /:/?,?: (fo p = 18a'*, to diMc ham so 3x^ + a '"^^ Ch{(ng minh rang day { x „ } co gi/li han Tinh giM han x^ + 25a4x 2x4 + i8a4 • ti ham, so ta thie.t ke ducic hoi loan 121 vd bai toan 122.^' Hu'cfiig dan Bai toan la mot t n t d n g hdp rieng ci'ia bai toan 117, vdi Xo = \/2 B a i t o a n 1 Gidi he phuong trmh I y = I z= - ^/W^ D a p so He co ba nghiem la ( ; 1; 1) (2; 2; 2), (-3; -3; - ) 302 25 • B a i t o a n Cho hai so thuc b vd a> Xet day so {xn}^^^ '"'^''^ , "^ = '= x„(x^ + 25a'') ^ ^„ ' ' \ 2x^ + 18a4 , V n = l , , u o , ^ : : Chiing minh rang vdi moi so thUc b, day so tren co gidi han hHu han vd tim gidi han 303 B a i t o a n 122 Gidi vd bien ludn (theo tham so a) he phuang 2x {i/ 2y{z' 2z{x* + 9a^) = y (7a^ - y^) + 9a^) = z(7a^-z^) + 9a'')=x{7a'' -x') trinh U'dng h d p a > c > K h i f{a) (1, ' '^ < f{c) =^3c<36=>c<6=^c=6=^a = 6=c ; ,, "ay moi tmtfing hrip t a don c6 a = = c = 1, suy = y/lO - 3x = v/4 - A / - 3x = X - 4.5.5 Svt d u n g h e l a p b a an d i s a n g t a c m o t s p h u ' d n g t r i n h he phu'dng t r i n h chiJa c a n V i d u Xudt phdt tic mot lie lap dcin gidn, ch&ng han Cach ,,,fjv Dieu kien 10 - 3x > { h-^ = A- <^ X = (tlioa dieu kien) \x < 3a ^ r 10 - 3x > ^ 90 - 27x-< 16 ^ 27 - ^ - 74 • ' < X 10 < ^ " T' Ta thay x = la mot nghiem ciia (1) Vdi diiu kien a, b, c khong dm thi he tren gidi a = b — c = Vdi x = 3, ta c6 diCdc di dang vd kit qud Id • Neu — < X < t h i ^ - 3v/10 - 3x < 3, k h i (x - 1) (x - 3) < => 4x - x^ > a = X - = 1, = v/10 - 3x = 1, \JA~ 3^10 - 3x "J JJ1 ;fb * / ij\;A 74 Vay — < X < khong thoa (1) Mat khdc, theo tren ta c6 I Neu < X < — t h i ^ - / ^ ^ > 3, k h i 4x - x^ = Ta thu diiac bai toan B a i t o a n 123 = 3c = 3\/4 - 36 = 3^/4 - 3^/10^^ - ' ' i ^ ''^' n'" " (x - 1) (x - 3) > =^ 4x - x^ < sau Gidi phuang trinh 4x - x^ = 3\/4 - S/TO - 3x (1) Vay < X < — khong thoa (1) Do x = la nghiem nhat ciia (1) Giai V i d u Xet C a c h Diou kion I f 10-3x^0 I 3v/10 - 3x < ^ r i - x > ^ \0 - 27x"< 16 a = x - , b = t v^lO - 3x > 0, c = \/A-3^/W^^ a > T i r (1) t a c6 he ( a^=4-3c \2 = - a c = " 36 ^ ^74 ^ ^ ^ ^ mot he lap ba an dan gidn : < = z + Q {thong thudng ta { z^ = x + Q nen chon cdc he cho vice gidi no Id khd dan gidn, vd chi c6 nghiem thod man dieu kien x = y = z) Tic he ta c6 10 J - > T i r dieu kien suv Ta dicac bai todn r 3a = f{b) , 36 = / ( c ) ( v d i / ( « ) = - i ^ , ^ > O) i 3c = / ( a ) H a m / nghjch bien tren [0; + 0 ) , v i /'(<) = -2t tong quat, gia sir a = max {a, 6, c} Trifcfng hdp a > > c K h i B a i t o a n 124 < 0, Vf > K h o n g mat t i n h < / ( c ) ^ a < = ^ a < = ^ a = 6=!>a = = c 304 • Giai.Datj sau Gidi phicang trinh v ^ = - ^ ta CO he j^hirdng t r i n h J x^ - A / G + s / a T + l = ^ { <-^^zl x^ = y + G y3 = + 2'^ = x + 305 (1, T i r (1), t a c6 x ^ - / - = Vay (2) (3) (4) (*) N h a n t h a y , lie (*) k h o n g t l i a y d i k h i h o a n v i v o n g q u a u h d o i v d i x,y,z k h o n g m a t t i n h t o n g q u a t , t a c6 t h g g i a t h i e t x = m a x {a;, y, z) n h a t t r o n g b a so x, y, z, h a y x > y, x > z) , y-^Q o • = x^ > y^ = z + Q^ y> , z=^y^ > _ i ne^ ( x l a so lot) N e u x > y t h i t i t (2) v a (3) n e n y ^ - , =^ - v a r 7^ - N h a n b a p h i t o n g t r i n h ( ) , (4) t a dUdc x y z = M a t k h a c , c o n g (2), (3), (4) t h e o ve t a d u d c suv = x^ + y2 + 2^ > ^xh/z'^ thco (3),(4) > z + Q> x + Q^ z> (3), J: x^y^z"^ < po d i n g thiic xay nen ; a:^ = y^ = z^ = I ^ D e n (lay t a g a p d i e u m a u t h u a n v(5i x > z V a y x — y K e t h d p v i (2) va (3) X = y = z = I (do X ; , , - , y ^ -I, z ^ -I) t a d u o c y — z V a y x = y = z P h i W n g t r i n h (1) t r d t h a n h p h i t d n g t r i n h co n g h i e m d u y n h a t x = - X - = 4=> (.X - 2) (x^ + x + 3) = <i=> x = " Bai t o a n 126 ( D e T h i l : l a i , n g h i o m n a y t h o a m a n b a i t o a n V a y p h i t r i n g t r i n h t r o n g dau b a i co n g h i e m d u y n h a t x = n g h i c h o k i t h i h o c s i n h g i o i c a c tru-ftng Chuyen k h u vvTc D u y e n H a i v a D o n g B a n g B a c B o n a m ) Gidi trinh X = y/S^.Vi-x + + V^-x.V5-x phudng v/S-xVS-x f x2 + X - I = y G i a i D i e u k i e n x < D a t s/S - x = a ^ v ' Vi du Xet ham la (]{t) = Kit Xri mM so f{t) he lap an ddn (jidn, clianq y2 _|_ = t'^ + t - Tren khodng - - ; +oo , ham Nghia la, tu z^ + z-1 = x, ta co z = hdp vdi he ta _ ^ _ i X = ngiCdc cua f{t) ^i±^±l^ Do d o 3-a^ i - P b - n g h i O l y m p i c / / 1 ) Gidi phuong G i a i Dieu kien x > a + b = trinh (1) • b + c = , d i e u k i e n z> ^ ^.6 ^ , — l a co 2z + = \/5 + 4x => 4^2 + 42 + = + 4x =^ = ab+bc + ca = ab + bc-\-ca =^ ab + bc + ca ^ = a.6 + 6.C + c.a ^' ( (a + b) ( c + ^ < (b + c){a + b)=i [ {c + a) {b + c) = e \ [ 2(2 r x(x+l)=:y+l y(?y+l) = + l + - = r + 30G =X + (2) r 2/15 a + b + c= v/T5 -—5 \/l5 + ——4 + yi5 G71 X = 240 Thi'r lai thay t h o i i m a n Suf dung can bac n cua s6 phiJc de sang tac giai he phifdng trinh C h o so p h i r r = r (cos^? + i sinv?), ?• > K h i d o cac c a n b a c n c u a z l a T a CO he p h i f d n g t r i n h f x2 + x ~ l = y \+ y - l ^ z ^ { - c + a = D a t y = x^ + x - T h a y vao (1), t a d i t d c Suy(• r a - + v/5 + 4x = - 6^ = - N h a n b a i ) h U d n g t r i n h t a d i t d c (o + 6) (6 + c) {c + a) = 2VT5 S u y r a ( x + x - l ) ^ + 2x^ + x = + y T ^ Dat z = - sau B a i toan 125 (De - x = b; \/5 - x = c, d i e u k i e n a, b, c l a so t h u c k h o n g a m T a co dUdc (.x^ + X - 1)^ + (.x^ + X - 1) - Ta CO bai toan < han = 0- cos , A; = , , , ? i - 1- + ism 307 va • Cac can bac hai ciia so phiic z = i- (cos ip + i sin < p ) , 7- > l a Tir day, ngitdc lai t a da t i m ditOc nghiem ciia he | ^^2y ^^^3 Z } zo=^ \/r (cos ^ + i sin ^ ) , zi = - v (|cos ^ + i sin , • Cac can bac ba ciia so phiic z = r (cos ip + i sin T = ^ c o s ^ , r > la f x - ^ c o s ^ y = ^ s i n ^ 20 = v / ? - ( c o s | + i s i n | j , n^Yi.rf3 ^ ^ feos ^ + , isin^i^) i[„h,, M o t phitdng t r i n h nghiem phiJc f{z) = 0, vcfi z = x + iy, t a bien doi h(x,y) + ifj{x,y) = 0<^ 'I y = ^ s i n ^ 'l ITTT y=v/2sin— 12 \/3 S a n g t a c c a c h e p h u ' d n g t r i n h b a n g e a c h l u y t h t e m o t so phiJc cho tru'dc Gia sit x + yi + -T'- Ta (/T/tfc 6di i o d n sau ^x'-Zxy'+{ixS-y')i = \ ^-^i , > 5 v / '"^"^ ^^•x-' + 3x^yz + x y ^ r + y^i"* = - + — i u | f^^^ ^xy^ = <^(x + y i ) ^ = Giai he t a t i m diMc x va y, t i t c6 z T u y nhien, c6 t h g t i m z bang each k h a i eSn bac ba ciia + i nhit sau : T a c6 + i = \/2 ^cos ^ + i sin (2) ' 2x^ - 6xy2 + (6x2y - 2y^)i = + bVSi + i<:>x^ + 2,x^yi + 3xy^z^ + y^i^ = + i ^x^ - 3x2/2 + (3x2y - y^)i = + | ^ ^ ^ _ ^ y I 5^3 G i a i N h a n h a i ve ciia (2) vdi i r o i cong vdi (1) t a dudc Do t i m can bac ba cua so phiic + i, t a t i m so phiic z = x + iy, x- G E , y € cho v/3\ (l + T^ Vay x + y i la can bac ba ciia so phiic = \ v/3 + V/ = (cos - 7r\ + sin-J M a z CO cac can bac ba la Vay cdc cfin bac ba ciia + i la • X zo = zi = V V2 VV2 (cos - + ^ sm - cos J= v^2 (cos - h I sin - — — f+47r 22 = Y V (cos ^ - y - h I sin f + 47r\ + z sin - = V2 Ztt = S/S ( c o s ^ + i s i n ^ ) , j, cos — + t sin — ITTT , ^ — j = V (cos 308 .• o'l) v/3 ' B a i t o a n Giai he phuang trinh (x + iyf ^l ^cos- i h(x,y) = \{x,y) = Nghia la m o t phirong t r i n h nghiem phiic, b i n g each tach phan thuc va phan ao luon CO t h e dita ve he phvtdng t r i n h 4.6.1 ' V i d u Xet so phiic z = (cos ^ + i sin ^ = V 3/ la so phiic thod man dieu kien (x + yif = = X = — ^ +I s m j zi = Vr [cos la 177r\ — + i sin S/5 , ( V / — Z2 ^ + 27r cos h i sin ? + 47r cos^^ + i sin f+27r\ — y f + 47r\ / 309 77r\ cos — + i sin — V 9 ; / 137r 137r^ N / cos — — h sm — \ , Vay cac nghieni ciia h§ phiMng t r i n h la X TT = 3/F cos — ^ 3/F ^TT .T = v^Scos .T = v o c o s — ' y=\/h cos - g da cho CO nghiem : y= 137r y = v^cos - x = x/2 cos dung t i n h (tang cap cua he, r o i dua ve phuong t r i n h da thufc bae ba iyY = + x^ + 4x^yi + Gx^fi^ + Axx/i^ + \fi^ = v^S + i K.: 4.6.2 x" - 6x2y2 + y^ = ^3 4(x'-*y - y^x) = = V^ ( ) x'-6xV+f B a i t o a n 128 Gidi he pfMOng trmh .r^y - y^x = - (2) z"* = (x + y i ) ' = x"* + 4x''yi + (Jx^i/P 6xV + 2/'') + (x^y - + 4xi/i^ D/ (cos - + t sm - ] , v/2 (^cos — + sin 37^ 257r 4/- / 257r V cos - + i sill 24 Vay z = x + ' 24 X yi ^ ^ ( c o s i L + yi x + yi= X + cos v/2 ( cos ^^^^ ' ''"'^80 — + i sin j 24 257r — + i sin 1377 24 7V5i z z ( zi + Z2 = I 2122 = ^ i + ^ •^x • ' •+ yi + <^x + 377r\ 24 X + y^ 7y/5iz zz zz 7y/Ei = H x2 + y2 x2 + y2 x2 + y2 7\/5x - 5y x2 + y2 rr~l x^+y^ y+ -7+ 5x + v ^ y + y^ = 7\/5x - 5y 5x + 7\/5y 5x + 7y5y X + X, = 7y5x - 5y x2 + y2 , /=0 =0 : a CO bdi todn sau 257r + I sin — 4/- / 377r 37?: yi = V I cos -r-r + I 24 24; 5z fc:#+ l ^ x + y + + '""24) 137r 310 24 • id sli z = X + yi, vdi x, y G R Khi phUdng trinh tren viet lai ,> Ha\ X + 3^ Vdy Z] vd Z2 Id nghiem cua phUdng trinh y^x) i phitc i; cos ^ + i sin ^ ) la b 24 zi = 7- VEi Z2 - s/hi { ,4,; ^ + ^'^"'i)' ^^ ' "^ la mot Ccln bac boii cua so plu'rc = ^ ssin i S a n g t a c c a c h e p h u ' d n g t r i n h tu" h a i s o p h i J c c h o t r i f d c + y^i = ^3 + ( = ^cos ^ + i sin nen tir he da cho t a c6 cos + z'^ ~7z + + 7V5i = ^ - + - G i a i X e t so phi'tc z = x + yi, vtJi x, y c M V i (x''- X = 1^2 24 377r V i d u Xet hai so phiCc z\ z^ nhu sau Ta CO hai Loan sau - 257r i;>'W;, <^ (a;'' - Gx^y^ + y'') + (x^y - y^x) i = \/2, + i V y = v/2sin 2^1 v^sin y = )/v^ f ta cd = ^cos- y = ^ s i n ^ cos — Lvfu y Cach gi^i rat doc dao va nhanh hdn nhieu so vdi each su V i d u Til{x-V 137r x = ^ c o s ^ X , a i t o a n 129 Gidi he phuang trinh < H bx + 7y/5y X 7^ = x2 + y2 7v/5x - 5y x^ + y^ 311 Hifdng dan Tit each sang tac he phuOng t r i n h ta thay he c6 hai nghieni 1^ ( X = y/5 y = x + B a i t o a n 130 Gidi he phUdng trinh 3x - x' ,21 y „ <2 iz == ixix - - y, + \^ — \zf = J? nghiem u + z.z y + i ^ = x^ + ir - ^ X'^ , + ^ = ^• \ VI+ 6it J (iv — \ + v^ J = 3\/2 + z u 3x-y , x2 + y2 x2 , .9^ + y^ 3(aH-yi) = X - yZ + y - xi X ^—+ y'^ST = —3-2 ^ -)- y 27i • 3^2 = V + ^ + ^ == < ^ !2= > + z +31 i_ + = ^ ^2 z.z z z + + i = ^I a2 - 62 ^ -3 f -3 ^' + y /1 + ^ V x + y ' r ^ i a i t o a n 131 Gidi he phuong trinh V ^/y Ta CO A = - 4(3 + i ) = - - i Xet so phiic a + bi thoa m an dieu kien ^ { \ CO bai todn sau = ^ ^Q - ' u = s/x, V = sjy, ta dU0c • lay - - 4i = (a + = — x + 3y + 1+ Lay (1) t r i t (2) theo ve t a dUdc ^ - y- + + '* = - i = 3\/2 + z = V trinh viet lai + v^ V (2) J •ifi + v'^ + 4=>u + (1) + 2i trinh cua phitany (3\/2 + i ) z + = < ^ z + - = 3\/2 + i <^ ^ + - He phitdng t i ' i n h da cho viet lai x Z (Jja sTi z — u + vi, phuong ^ Xet so philc z = x + i y K h i _ va(I 22 ^(i - +y x^ + G i a i Dieu kien x? Vi du Xe'^ hai so phiic | =0 .T x + = V2 y ) ^ « B a i t o a n 132 ( H S G Q u o c g i a n a m h o c 0 - 0 ) Gidi he trinh phuang K h i Vay (a; - = - <^ a"* + 30^ - = ^ (1; - ) ; = a = ±1 (a; /;) = ( - ; ) Do A = - - 4i c6 hai can bac hai la ± ( - 22) Suy y + 3x iai Dieu kien x > 0, y > 0, y + 3x 7^ Dat u = \/3x > 0, u = hay vao he, t a diTdc 1 + 12 n2 + i;2 ; 12 73 V = u— = Q V + 12u + v' I2v Nghia la X - yi = X - yi = 1+ i ^ i x;y| = f ; l ) x;y) = ( l ; - l ) Vay he phudng t r i n h da cho c6 hai nghiein la | ^ Z ^ 312 > • + v^ = 2v/3 (1) = (2) ban phitdng t r i n h (2) v6i i , sau cong vdi phurtng t r i n h (1) t a ditdc va r x = y = -l u + vi \ = 2v/3 + i 313 (3)' / Xet, so phi'rc z — u + vi, v6i u > 0, v > K l i i (3) viet lai - i + S Ta V ^ z.z £ = B a i t o a n 134 Giai he phUdng 12 z - ^ = 2(v/3 + 2\/3 + 6i = ^ ^ - ( v / + - 12 = (4) CO A' = {V3 + 3if + 12= - + 6v/3/ + 12 = + oVSi = 12 Vay A ' ' CO 2+ T i t (4) t a + M » y t = + xy trinh G i a i Dieu kien x - , y 7^ - He treu t u y la doi x i i n g loai I nhirug bac kha cao, vice dita vc = x + y va P = x y c6 the gap nhieu kho khan Nhirng neu t a biet bien d o i t h i bai toan t r d nen ddn gian Tritdc het t a chiing m i n h (1) T a c6 - ^ + ^ y + \/3 = ± (3 + v ^ i ) ^ + 2^ = ±/l2 <»x^ + X + X = l ^ x { x + l ) + y{y+l) = {x + • l){y+l) ^ + y^ + y = x y + X + y+ ' > x''^ - x y + y^ = Vay (1) dung D a t — ^ = a, -^^r^ ' ^ phitdng t r i n h m6\ CO = v/3 + 3/: + + v ^ i = v/3 + + (3 + \/3)« = \/3 + 32 - - v/3i = v/3 - + (3 - v/3) Do ti > va V > nen u= v/3 + v = N/3 + [a^ ^l2a2-2a = ^ + b'' = l 2/= (v/3 + 3) = + 273 y = (4 + 2v/3) B a i t o a n 3 ( H S G q u o c g i a - 9 ) Giai he phmng ^3x1 + T + ( 1- ^ \ y X trinh ^ + y + xy = 5y + (1) , x+ „ , = ^ l + (x + l ) ( y + 2) ; (2) Y tifdng T a c6 (1) <i=^ (x + l ) ( y + 1) = C) (2) cung c6 x + va y + nen t a nghi t d i dat a = x + 1; = y + T u y nhien m a u chot ciia bai toan chinh trinh = ) = 4x/2 la dat c = ^ de tvr (1) c6 a6c = K h i t a lai c6 menh de sau : "-^""-^^ l+a + ab^ l + b + bc^ Phu'dng p h a p b i e n d o i d a n g thu'c Nhieu bai he phitdng t r i n h t u y nhin phi'rc tap nhimg c6 the giai bang nhGng dang thiic ddn gian M a u chot giai nhitng bai toan dang l a t a phai nhin l a quan he giifa cac an so, tvr lap nen nhirng hang dang thirc t h i c h hdp M o t so dang tln'rc r o t h g khong quen thuoc, nen phUdng phap doi hoi k i n h nghicni va sir t i n h y _ ^ l+c+ca T i t t a d i den Idi giai G i a i Dieu kien x 7^ - G , y 7^ 314 [ ( a ; = ( ; 0)- Tir t i m r a nghiem ciia he l a (x; y) = (0; 1); (x; y) = ( ; 0) B a i t o a n 135 Giai he phmng He da cho c6 nghiem nliat | ^ ~ ( + ^ ) 4.7 I + = 12(cos- + s m - j hai can bac hai la ± \ / l ( c o s ^ +ism'^^ X D a t x + = a, y + = 6, - = c T i t (1) t a CO a?>c = va + a + a6 = xy + 2x + y + = + (x + l ) ( y + 2), y + 5y + , l x + ^+6 l + + 6c = l + y + l + ^ ^ = l + c + ca = + - + - J - = Ta CO he phUdng t r i n h { abc = l + a + ab ; 1+h + br 315 1+c + ca VSi abc = 1, ta c6 ^.8 1 + + + a + ab + b+hc l + c+ca + a + a6 l a ab = 1 + a + ab'^ a + ab + abc^ ab+ abc + a?bc l + a + ab x+ <^x = l , d o d t / = l Vay he c6 Thay vao phitdng trinh sau dUdc = + he phmng xy = + 15x2/(x + y) = 32 + Y tifdng Ta can tim moi quan he giifa cac hang t i i he Muon vay ta chii y tai hang dang thiic (x + yf ^ + y^ + bxy[x + y){x^ + xj/ + y^) Giai He da cho duoc viet lai nhu sau: + yl + xy = 2> \x'5 + 2/^ + 5.3xy(x+y) = 32 ' {xl : ^ / x + y2^xy = ^ + iy5 + 5.Ty(.x2 + r y + r x^ + y = z + y = z + y = ^ ( x \ + y) = xy ^ \ z ^ C x (x + = xy B a i t o a n 137 Giai he phUdng + y) = y)2 = 1 <=>^ liJ (1) x'' Bai t o a n 139 Gidt he phuang trinh | / + 2/^ + + y^ + yz - (x + y ) - ( x + y ) + 2 - y ) - zix + = Dat H = X + y , I' = X - y =j> X = (.) x y z trlnh y"' xz z^ xy 1\ z^j (2) z^ 2/2 = 0^ / feW 'Jf^ =^x•^y = ^ = + 2xy - zx - 2y = 2xy = - zx - X ^ 316 ^' z^ -(^ X T I v ; ^'^^ (2) CO nghiem va chi -''i i P " '' he da cho CO nghiem la (1;0;2), ( - ; ; - ) f \xy - 18| = 12 •^1 t o a n 140 Giai he phuang trinh I _9_|_1 * -2 ' + = • Vdi = - thay vao (2) ta co w = t; = - Suy { yZQ^ % = y =: ' He (1) trd y= _ = •^61 = thay vao (2) ta c6 « = v = Suy | y = Q z z The vao he ta tim dudc nghiem (x; y; 2) = .Srt6t/.t f \ IV - + - + = \x y z) xy 12 —2 + —2 = yz zx xy z'^ 1\ = Vy 1\ / I 1\ zj +1 - + - -+- Vx = 4=» + - ^ + - ^ + —2 + [x^ /I 2/ xy M i « Den day ta gap mau thuan vdi dieu kien Vay he vo nghiem 32 Giai Dieu kien x ^ Q,y ^ 0, z ^ Ta, c6 : 1\ z ''^'^ Giai He da cho turtng dirdng xy + y = ' i + i + i = - + - + Vx y ;x • ^|(,''+|)V^=o =^{x2';y+xy = \ Vay he c6 nghiem nhat (x; y) = (1; 1) /I *x + y i z- \x - (1) ^ 'rh Cfiai Dieu kien xyz 7^ Khi he da cho vi6t lai ( x irinh so he khong m a u m L f c a i t o a n 138 Giai he phuang trinh nghiem nhat la (x; y) = (1; 1) B a i toan 136 Giai Mot x2 Qiai Ta co |.Ty - 18| = 12 - x^ =^ 12 - xy^-d+^y"^^ \x\ 2^3 |y| 317 .T^ > ^ |x| < v ^ |x| > 2v/3 '•' • Hifo'ng d i n Ta CO (1) o 4x (2x2y - 2x2) _ Suy | x | = 2s/3 => xy = 18 K h i x = -2\/Z t a c6 y = -ZVZ K h i x = 2y/^ ta CO y = \ ^ Thilf lai, thoa m a n he da cho Vay nghiein cua he la ^ 8.r3 (y - 1) - (4x + 1) (y - 1) = \ / f e + T ( y - 1) y (x; y) e { ( - v ^ ; - v / ) , ( ^ ; ^ ) } + sJ2x + y + 2xy + = S/S^in: = 8x3 - 2y - X > = ! i',;,- (1) B a i t o a n ( D e n g h i cho k y t h i h o c s i n h gioi c a c truTcJng C h u y e n k h u vu'c D u y e n H a i v a D o n g B a n g B a c B o n a m ) Gidi he-phudng trinh i- (2) (x - y) (x2 + x y + y2 - 2) = In y+_v^2^ (1) x''y — 3xy — = G i a i T a c6 (1) ^ (2x + 1) - (y + 1) + v/(2x + 1) (y + 1) = I (1) ^ (x - y) (x2 + x y + y2 - 2) = in =0 „3 X"^ ^ (2) G i a i T a c6 Dieu kien (2x + l ) ( y + 1) > M a x > nen y + > B d i vay (1) <^ (\/27TT - v / y + l ) ( \ / x + + vVn) ,.; 8x3 _ 4^ _ = ^ x + B a i t o a n ( D e t h i h o c s i n h gioi c a c tru-dng C h u y e n k h u vuc D u y e n H a i v a D o n g B a n g B a c B o n a m ) Gidi he 2x-2y _^ J) ^ s/6^TT (y - 1) ^ J) ^ \/2x + - -v/y + = <^ y = 2x J/+ ^2/2+9 \^x + y ; ; ^ : r / T^M^) - 2x + 61n ( x + v/x2 + ) = y3 - 2y + 61n ( y + (*) :et ham so / {t) = -2t + G In + Vt^Td^, « € M Ta c6 T h a y vao (2) t a diWc ^ x + = 8x3 - 4a: - (g^ j^Y)^ ^ x + = (2x)^ + 2x /' (/) = G -2 + = 3/2 + + v / ^ 3;- ' H a m so f{t) =^t^ ^-t dong bien tren M Do ,c6 (3) <^ x/feTT = 2x 4x3 _ 3x = i (4) De thay rftng x > khong la nghiem ciia phirdng t r i n h X e t < x < Dat X = cos a, vdi < Q < - T i t (4) t a co V72T9 <2 + 27 ^ v / ^ ^ 29 26 29 / ^ ^ 27 29 , 26 29 29 >1+ = - 3 29 „ =0 fSuy / ' (0 > 0, V< e M Vay ham s6 / dong bir-n va lien t n r tron L M a W y he c6 nghiem (x; y) = ^cos ^ ; cos B a i toan 142 ( D e nghi O l y m p i c / / 1 ) 4x (2x2y - 2x2 + 1) _ - 3x = ^ (*) ^ Do < a < ^ nen " = „ + v/t2T9 + v , 26 , «N > + — ( r + 9) - ^ ' ' cos 3a — ^ Gidi he phMng (43 ^_ i ) + j ^ ^ / ^ q n : ( y - 1) (1) (2) 318 trinh / (•^•) = fiy)^^ = y- fj I T h a y vao phiWng t r h i h (2) ci'ia he ta ('6 x*^ - 3x2 _ = (3) ' Dat x2 = u, dieu kien u > K h i u3 - 3u = 319 (4) Dat u = 211 (dieu kien v > 0), thay vao (4) t a dUdc 4ir-3v = Vdi - (5) 9 v/3x + + (x + 0; (x;y) = , V = COS • „ Do dieu kien v > nen chi nhan i ; — cos ^ Tit x = ±./2cos - He c6 hai nghiem la x^ - y^ + if_ G i a i Digu kien { Gidi he phmng (2012 - 3x) y / I ^ + (6y - 2009) y / S ^ ^ = (1) 2^/7.7; - 8?y + 3V/14.T - 18?/ = r^ + 6.T + 13 (2) dUdc \&\ = 1\ , (x;?y) = ( - l ; - l ) 2; - 3x -2 = (1) x2 + y r ^ ^ - v / 2 r ^ + m = (2) -W2cos^;-y^co^^ B a i toan 144 ( D e nghi Olympic 30/04/2012) — + > 0, vay (5) cho t a v/Sx + + (x + 3) B a i t o a n Tnn cac gid Iri thuc ciia m de he phuong trlnh sau c6 nghiem , 2cos|; j c o s ^ 2) + cac nghiem x = 0, x = - Vdi x = t a duoc y = J 7n = COS thi y = - T h i i lai t a dUdc cac nghiem ciia he phUdng t r i n h da cho la Theo ket qua bai toan d trang 119, suy cac nghiem ciia (5) la V = COS—,V X < tnnh ijf/^^ ^ { 0<| V.'- , ,^ < ' ^ ^° (1) ^ x^ - x = ( / - ) ^ - ( / - ) (3) x2+ \ / r ^ - \ / r ^ + m = < ^ m = \ / r ^ - x Xet ham so f{t) (3) Xet h a m so g{t) = (3i + 2000) ^Tt, V( > K h i suy h a m so / dong bien tren [0; + 0 ) D o /(4 - x ) = /(3 - 22/) ^ - X = - 22/ <^ y = (4) Thay (4) vao (2) t a dUdc 2sjlx - t, Vi e [0; 1] V i g'{t) = -1 - < x^ + v / l - - 3y/2y - 2v/3x + - (x + 2)] + [3v/5x + - (x + 3)] = V x + y + V2x + y = V'2x + y + x - y = + x G i a i Dieu kien m i n {7x, x } > -y D a t s/lx + y = a va v/2x + y = h TiT he -2x(x+l) -3x(x + l) H — 7 = x ( x + 1) v / x T + (x + 2) \ / 5— ^ ^ ^+ (x + 3) v/3x + + (x + 2) + 320 v'Sx + + (x + 3) Gidi he phuang trinh <=i>2\/3x + + 3v/5x + = x^ + 6x + 13 - < nen < B a i t o a n 146 ( H S G Qu6c g i a n a m h o c 0 - 0 ) - 4(x - 1) + 3v/l4x - 9(x - 1) = x^ + 6x + 13 <^x(x + 1) = 2vT^ (4) ham g lien tuc va nghich bien tren [0; 1], suy tap gia t r i cua h a m g la [t/(l); g{0)] = [ - ; 2] Vay he da cho c6 nghiem 4^ phUdng t r i n h (4) c6 nghiem tren doan [ - ; 1], ti'rc la m thuoc tap gia t r i cua h a m g, hay - < m < L\iu y Tfr bai toan tron t a t h u ditdc ket qua : Neu hai so thitc x G [ - ; 1] va y G [0; 2] thoa m a n dieu kien - y^ + 3y^ - 3x - = t h i , / ( „ = 3v/;+5i±|lL»>o,v,>o (3) , l a m so f{t) = t'^ - 3t c6 f'{t) = 3*^ - < 0, G [ - ; 1] nen / lien tuc va " g h i c h bien tren doan [ - ; 1] V i vay, t i f - < x < 1, - < 2/ - < va tir (3) t a C O / ( x ) = f{y -l)<^x = y- l-i^y = x + l Thay vao (2) dUdc G i a i Dieu kien x < 4, 2/ < 2' 7x - 8?y > 0, 14x- - ISy > T a c6 (1) ^ [3(4 - x ) + 2000] \ / ^ = [3(3 - 2?/) + 2000] sJZ - 2y ' • + = phUdng t r m h da cho t a C O he (5) a + 6= (1 + x - y = (2 321 Nhan thay - 6^ = 5x Do 4=> cos'^ {a + h){a-b) K i t hdp (3) vdi (1) suy - = 5x'^"=i'\-b h = X ^, = x (3) the vao (2) ta ditdc A C B ^ + cos'' ~ + cos^ —- = ^ <^ cos A + cos B + cos C = ^ <^cosyl + cos/3 +cos(7r - (/I + S)) = + x - 2/ = < » x = 2 / - l (4) The (4) vao phiTctng trinh thii hai cua he ta dUdc XI + X2 a;20i3 + 2:2014 = a;f 14 X20U = 74 ^ "'2"' V " ^ I' Y T = C 7' ^ ' ^ i 0<A,B,C<7r Thay 2X > X2"14 ^ > X^"!'^ (vi X >0) j Af B C\ C A - tan —tan — ^ tan - (^tan - + tan ^ = ^ " ^^n - tan ^ ^ tan ^ = ^-2 C -1 tan — + tan — 2 + _ = _ _ _ + ^ ^ ^ ^ + 2? + C = 7r + ^2^ Do dieu kien < / I , Z^, C < TT ta c6 ^ + Z? + C = TT PhifOng trinh (2) trd •+ l + tan2^ ' l + tan^l + ' + tan^ | 322 ^ (2)- Tft (1) va (2) suy X^oi^ = ^2013 = Ngliia la ta c6 X I = X2 = B (1) Lap luan mot each titdng tit ta cung di den < y2013 , B B C C A tan - tan - + tan - tan - + tan - tan - = 1 Lai ton tai k e {1,2, ,2014} cho xk = X nen suy vao (3) ta dudc <^tan- = cot(^- + - J ^ - jt; X > xfi4_VA;= 1,2, , 2014 (2) (3) f ;: ; • V, Titdng tit doi vdi cac phudng trinh ciia he ta c6 : (1) *„ „ • V T " , L X,=xf'^ 2X > X I + X = x f i Hifdng dan Do x, y, z difcJng nen Hut + Giai Gia si't ( x i ; x ; ;x2oi4) la nghiem ciia he Goi X la gia t r i Idn nhat ciia cac so X j , i = 1, , 2014 va Y la gia t r i be nhat cua chung The thi tit phUUng trinh dan ta co : ,, , B a i toan 147 Tim cdc so thuc dudng x, y, z thoa man he phuong trinh + yz ' y + zx ' z + xy = x f 14 x f 1" X2 + X3 = Bai toan 148 Giai he phuong trinh : \ K X ,,ti:i;= 4=> cos >1 + COS B - cos(yl + i?) = ^ - ^ y l = B = C = ^ (do bai toan 161 d trang 111) Vay x = y = = -1 j , v / y - + y - l = ^ v/52/ - = - y ^ / / < 5y - = - 62/ + y2 J y <3 ^ 11-^77 I -11?/ + 1 = = —2—• S: / The vao (4) suy nghiem cua he la (x; y) = I 10 - v/77; n-V77\ •' - • ''"^ ' Thit lai, ta ket luan : (xj; nhat ciia he ••• = X20H X2;X2014) = = ( ""v/2; Bai toan 149 Giai he phuang trinh { ^ (v^+y v2 '"'s/2; ; '"'s/2) la W^^) 1^ v/x + y + V x = G i a i Dieu kien x > 0, x + y > (1) titdng ditdng vdi :v {s/TTYi + s/TT^ X + - V nghiem (1) = ( T + y) - (x + 3) ^ x = ^/T+Ti - Vx + ay (2) trft (3) dildc Vx + = ^ X H / + 2\/X2 + X = 323 (2) (3) PhUdng t r i n h t h i l nhat ciia he viet lai 2x-l)f2y-iUl Vay (x; y) = ( ; 8), thoa man dieu kien B a i t o a n Gidi h$ phudng trlnh + l=4xy \ = (1) \ ( l ) < ^ x + \ / l + x2 = V ' H - / - y < ^ a ; + j + \ / l + <^ x + y + — = fix) = > /(2) = Bai toan trinh X + y = Khi X > 0, t a CO /3 ^ + \ x2 W- + 2+ = -4 + - + V^ x'^ x ^JtVi = t-\ K h i X < 0, ta CO - J- 152 (China \/u + = u \ \ + - + 4T X X - ^/ V yJ > - Girls M a t h OIympiad-2005) 1\ X + - Gidi he phudng + - (1) (2) ( I ) ^ ( x + l ) ^ ( y + ) ^ ( 2 + 1) -4 x^ x2 (2) dan den T i r (2) suy X , y, X X'^ - B a i t o a n G i d i he phicang trlnh [ V 2y - G i a i Dieu kien x 7^ 0, y 7^ 0, z 7^ T a c6 X^ Dat i = - + 2x - - / 1\ 1\ = 12, y + =13 \ xy + yz + 2x = 1.\J / + + ^ = - + - + -^ Dat u = - + V dan don - = x2 f-4 / ( y ) > / ( I ) = =^ Vay (3) <^ (x; y) = (2; 1) va la nghiem nhat ciia he T l i a y 2/ = - X vao (2) diMc x\/3x + x + = - x + 3^ + ^ ^ X , x2->/l+y2=Q (.T + y) ( V l + x2 + V T T y ^ + X - (3) tang Do v6i x > va y > t h i H u - d n g d i n Dieu kien 3x - 2xy + > Vi t + Vl + t^ > nen x'^ - 7/2 = — VliV X yJ = i - - , vdi t > De dang chiing m i n h ditoc / la ham Xet h a m so f{t) (2) + 3x+l ; A ,), |, svYr^ir^ + v / T T ^ ) (y + y r r ^ ) f i ciing dau Ngoai ra, neu (x; y; z) la nghiem ciia he t h i ( - x ; - y ; - ) cung la nghiem ciia he, nhir vay t a chi can t i m cac nghiem ditong T i r (1), suy t o n t a i a, /?, € (0; TT) cho (^^^ " l ) (2y^ - l ) = ^^^y a + /3 + = 7r, a /3 x = t a n - , y = tan 2^ ^ = tan- [ x2 + y2 + x y - 7x - 6y + 14 = T h a y vao (2) t a ditdc G i a i Phitdng t r i n h thiJ hai ciia h f viet lai x2 + ( y - ) x + y - y + 14 = ( l + tan2^) 5(1 (1) Dieu kien de (1) c6 nghiem la tan- (y - 7)'^ - (y2 - 6y + 14) > 3y2 - lOy + < ^ < y < - sin a y2 + (x - 6)y + x2 - 7x + 14 - (2) sin ft 324 X < — tan- sin (3) Theo d i n h l i ham so sin, t h i tir (3) suy a, (i, la ba goc ciia mot t a m giik" CO dai cac canh tUdng iltng la 5, 12, 13 T a m giac la t a n i giac vuong =-, 3x2 - 16x + 20 < <^ < 13 ^1 + tan2 ^ ) 13 CO Di^n kien d^ (2) c6 nghiem la + tan2 tan- 12 Phirong t r i n h t h i i hai cua hO viet lai (x - 6)2 - (.x2 - 7x + 14) > 12 ^ 12 sniQ = — , sm/3 = — 13 13' 325 , • K l i i x^ = y"' <=> a- = t/, thay vac ( ) , t a dvWc sin Q Ta CO = 2t l + 19 1+ x^ - X = « • X = ^ fdo dang xet x 7^ 1O 13 •H-'""' Vay ( x : y ) = (^g^ g j a TT -K ,^ rt , ^ < a < - = > < - < - = i > ( ) < tan ^ < 2 i Dat (V A O tan 77 = F • ' ^ i " l O t n g h i e r n c i i a he ^ Khi x ' + = - ( I l i a p h u o n g t i i n h (1) c h o x c h i a phUcJng t r i n h (2) c h o y , (hrri< ^:^^ + ^ - y ^ - ^ = 2v 12 2v '{[ o'J = t a n ^ K h i ; •' (5) 12i;'^-2G« + 12 = = ^ u G , ' png (5) va (6) ta t h u diwc Taco y < / < ^ = > < ^ < = ^ < tan ^ < => tan ^ = ^ 2 z X y (7) xy V a y cac nghieni ciia he la , , l a x ^ + y ' = - l i e n t h a y v a c ( ) , t a d u d c xy ^ - -i^ x ^ y ^ = - N h i f v a y x^ /1 \ \o J / \ \ B a i t o a n 153 ( L i t h u a n i a n M a t h e m a t i c a l phuang X va y ' la nghieni ciia phitOng t r i n h J / Olympiad 2006) r Giai he ., + - = <^ 8/,- ^ 9< + = o Irinh -xy' -\x 1, (i) = {) § y4 + ^.2 „ _ (*) = - ^x^ = Z + xV-yV-fy'^o t a dm.K: " 8' 0'-:y)-(l;^) U-;.v) = ( x ; ) , v i X- ^ va ( ; y), v(n y 7^ T i e p t h e o , gia s u r 7^ va y 7^ N h a n pliUdng t i i i i l i (1) vdi^x^ v a nhan phUdng t i i n h (2) v i - x'y' (2) G i a i D t h i y {x\y) - ( ; ) la nghieni ci'ia he va he khong c nghieni dang x« + Do / = _ (^;i) j i j C a c u g l i i e i i i e i i a he l a ( x ; y ) = ( ; ) , ( x ; y ) = ( g ; ^ ) , ( x ; y ) = (1; ,1 ( x ; y ) = ( i ; 1) (3) (4) L a y p l n t d n g t i i n l i (3) tnx p h i l d n g t i i n h (4) ve t l i c o ve, t a t h u d u d c <',X, (f:),i y' = l { x ' - y ' ) ^ { x ' - y ' ) { ? ^ y ' - l ) = () • x ^ - y-^ = x^ + 2y^ = 326 327